register_taxonomy for both ‘Post’ & ‘Page’

The second parameter of register_taxonomy can accept a string or an array of post types, so simply add the page post type to that array, like so:

function qc22_build_taxonomies() {
    register_taxonomy( "fullscreen-tags", array( "fullscreen", "page" ), array( "hierarchical" => true, "label" => __( "Fullscreen Tags" ), "singular_label" => __( "Skill Type" ), "rewrite" => array('slug' => 'fullscreen-tags', 'hierarchical' => true ) ) ); 
}