Is there a way to get an overview of all skills among all my dwarfs, either in vanilla DF or with a tool?
Dwarf Therapist is an external tool that lets you manage job assignments, gives you statistics, and lets you sort your list of dwarves by various criteria, including by skill.
This is a screenshot of a fortress population sorted by Farming skill, with the degree of skill indicated by the size of the box (or a diamond for legendary skill):
It accomplishes this by reading the data from the memory allocated to Dwarf Fortress, so it has to be matched to your version of DF. Thankfully, it's possible to update its memory mapping for new versions of DF without waiting for a whole new version of Dwarf Therapist to be made available.
It's available for Windows primarily, with an experimental Mac OS X version also available.