If you use .* as "everything else", then be sure to use the /s modifier.
s/^.*?(\d+).*$/$1/s
Beware of that if there is no digit in $file, then $sk won't change! This is bad. It would look like you need to change \d+ to \d*, but that would make it match "" at the beginning, and so $1 would be empty, thus erasing the whole string.
But this problem can be solved. By rewriting the pattern to a more natural (?) pattern we'll soon see the solution. First, your pattern can be rewritten to
s/\D*(\d+).*/$1/s
The anchors are removed, as they're unnecessary. They're unnecessary in your pattern too. Anyhow, now we can change \d+ to \d*, and $sk will be empty if no number was matched. So the result is
(my $sk = $file) =~ s/\D*(\d*).*/$1/s;
But this still isn't fully analogous with
my ($sk) = $file =~ /(\d+)/;
since $sk will be the empty string in the former, and the undefined value in the latter. So in extraction situations I often stay away from that trick, and simply use the latter.
ihb 