How to write a new file with output from script

We need more details to give you an exact answer.  In general, Perl code would look something like this:

use POSIX;
open OUT, ">file" or die $!;
# do remote stuff and capture output (backticks, open pipe, etc)
print OUT $name, strftime('fmt_you_want', localtime), $remote_results;

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Here is a example of my perl script.
Where would I place you example in this code?

#!/usr/bin/perl                                                                                                                                
use strict;                                                                                
use warnings;                                                                              
use Net::OpenSSH;                                                                          
$|=1;                                                                                      
my $user = 'root';                                                                                                                                        
my $password = 'rootroot';                                                                 
use vars qw ($cmd $host @HOSTS @CMDS); #                                                   
#                                                                                          
my @HOSTS= (                                                                               
                "server1",                                                       
                "server2",                                                       
                "server3",                                                                           
);                                                                                         
my @CMDS= (                                                                                
             "hostname",                                                                                                                     
             "service rsyslog status",                                                     
             "df -h",                                                                  
             "echo '***** *****'",                                                     
);                                                                                         
                                                                                           
foreach $host (@HOSTS[0..1]) {                                                             
chomp $host;                                                                               
#print "$host\n";                                                                          
my $ssh = Net::OpenSSH->new(host=>"$host", user=>"$user", port=>22, password=>"$password");
  foreach $cmd (@CMDS) {                                                                   
   chomp $cmd;                                                                             
   #print "$cmd\n";                                                                        
  $ssh->system($cmd);                                                                      
  }                                                                                        
}  

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You'll have to change your ssh call.  This should work:

#!/usr/bin/perl
use strict;
use warnings;
use POSIX qw(strftime);
use Net::OpenSSH;
$|=1;

my $user = 'root';
my $password = 'rootroot';
use vars qw ($cmd $host @HOSTS @CMDS); #

my @HOSTS= (
                "server1",
                "server2",
                "server3",
);
my @CMDS= (
             "hostname",
             "service rsyslog status",
             "df -h",
             "echo '***** *****'",
);

open OUT, ">>ssh_out.log" or die "could not write ssh_out.log for $cmd: $!";

foreach $host (@HOSTS[0..1]) {
  chomp $host;
  #print "$host\n";
  my $ssh = Net::OpenSSH->new(host=>"$host", user=>"$user", port=>22, password=>"$password");
  $ssh->error and die "failed to connect to $host: " . $ssh->error;
  foreach $cmd (@CMDS) {
    chomp $cmd;
    #print "$cmd\n";
    my ($rout, $pid) = $ssh->pipe_out($cmd) or die "$cmd failed: " . $ssh->error;
    print OUT join('  ', $host, strftime('%Y-%m-%d  %H:%M:%S', localtime), $cmd), ":\n";
    while (<$rout>) {
      print OUT $_;
    }
  }
}

close OUT;

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I was able to get that to work but how would I go through a 2nd loop an copy it to same log file?

open OUT, ">>ssh_out.log" or die "could not write ssh_out.log for $cmd: $!";

foreach $host (@HOSTS[0..1]) {
  chomp $host;
  print "$host\n";
  my $ssh = Net::OpenSSH->new(host=>"$host", user=>"$user", port=>22, password=>"$password");
  $ssh->error and die "failed to connect to $host: " . $ssh->error;
  foreach $cmd (@CMDS) {
    chomp $cmd;
    #print "$cmd\n";
    my ($rout, $pid) = $ssh->pipe_out($cmd) or die "$cmd failed: " . $ssh->error;
    print OUT join('  ', $host, strftime('%Y-%m-%d  %H:%M:%S', localtime), $cmd), ":\n";
    while (<$rout>) {
      print OUT $_;
    }
  }
}

foreach $host (@HOSTS[2..9]) {
chomp $host;
#print "$host\n";
my $ssh = Net::OpenSSH->new(host=>"$host", user=>"$user", port=>22, password=>"$password");
$ssh->error and die "failed to connect to $host: " . $ssh->error;
  foreach $cmd (@CMDS[0..2, 9..10]) {
   chomp $cmd;
    my ($rout, $pid) = $ssh->pipe_out($cmd) or die "$cmd failed: " . $ssh->error;
    print OUT join('  ', $host, strftime('%Y-%m-%d  %H:%M:%S', localtime), $cmd), ":\n";
    while (<$rout>) {
      print OUT $_;
  }
 }
}
close OUT;

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What do you mean?  The script appends to the existing log (>> instead of > in the open) and only opens it once for each run of the script (open is outside of all loops).  Basically logic looks like this:

open log file (append)
    connect to each host
        run each command on each host
            write output to log file (for each command on each host)

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Sorry about last question but I understand., had a type issue.
         
                   How could I put same strtime on the log name?
                   Is it also possible to see the results on my screen as it getting copy to log file?

One more question, How would I place the logs in a specific directory?

Wow, never thought about turning that HOSTS array into a hash.
Thanks works well.

Thanks!