Just change two lines:
add the $page
variable and then update the value with it. So your foreach
postlist should be:
<?php
$postlist = get_posts( 'posts_per_page=-1&post_type=skills' );
$posts = array();
foreach ( $postlist as $post ) :
$page = get_page_link( $post->ID );
$option = '<option value="' . $page . '"' . selected($page, $current_page, false) . '>';
$option .= $post->post_title;
$option .= '</option>';
echo $option;
endforeach;
?>