As someone mentioned, FATE/Fudge is heavily skill-based, though you may want to find either a worldbook or a game that runs on the Fate system to get an idea about how that works. Granted, for FATE, the skill system can be loosened up enough so that essentially is a stats based system, or run as a skills-based system. (The Fate Freeport Companion essentially does this to model D&D fantasy -- you have six skills that match up to the D&D six attributes; meanwhile the Dresden Files RPG keeps more traditional skills, but still has the magic-casting fun time.)

The Planet Mercenary RPG uses skills as its core 'this is what your characters can do', but I don't know how you feel about a media tie-in RPG when you don't know the media in question. Anyway, it's comedy space-opera about playing officers in a mercenary company. (Players pick a race, a 'command package' (what they do for the company) and a background that all give ranks in either specific skills or 'pick one from this group', and then get extras to customize the character. Advancement is done via adding skills. Races can also give skill penalties and a few abilities that aren't modeled by skills. For instance, uplifted elephants get a few skills (Athletics, Endurance, Perception, a free skill speciality (Athletics: Run), and a penalty to Dodge because elephants are not known for being small and nimble, and the note that their trunks can act as a third hand with limited dexterity).)