Skins are part of the skeleton data, they don't go in an atlas. You can pack the images for the attachments in each skin in separate atlases, but at runtime you'll need to know which atlas to look in to find the region for each attachment.
An alternative is to pack the images for each skin in the same atlas, but on a separate atlas page. You can use folders to decide which images go on their own page:
Texture Packing - Spine User Guide: Folder structure