Is there a table top game similar to the video game Dungeon Siege where you don't really pick a class but what you become depends on what you do? You cast magic your magic skill increases, you beat someone with a stick your fighting skill increases that sort of thing.
The original Runequest was a lot like that: the world was quite magical, and most PCs had some magic available to them. Experience worked just like you're thinking about. When you used a skill, you checked it off, and if you could roll higher than that skill during the downtime, it'd improve. It was thus easier to get better at a skill the worse you were at it. Some of the other Chaosium games based on the Runequest system share the same method.
Mongoose has recently released Runequest II, which shares the same advancement mechanic.
You could do a variant of GURPS that did the same thing pretty easily, basically by requiring that a person has used skills in order to improve them (or you could track their improvement yourself). EABA would work similarly, and EABA doesn't encourage the glut of skills that GURPS tends do.
Or perhaps a PDQ variant, you could do the skills similar to the way Soul Cultivation works in the Dead Inside variant, or Hero Point gains in Truth and Justice.
A lot of this depends on the general feel of the game. For something pretty light, I'd tend toward PDQ. If you want a bit more crunch, EABA or GURPS could be made to work, I think.
Ars Magica has a separate XP track for every skill and Art. You're only permitted to increase these if you've used them during an adventure or if you explicitly spend time (as in, a season or more) working on them during down time. That seems to fit your criteria.
Star Wars d6 and the original Cyberpunk 2020 required the player to use the skill (or be trained in it by someone else) in order to improve them.
Note: Cyberpunk had classes (which only effect meant that they got one class specific special skill) while Star Wars d6 had character templates that didn't have any concept of class at all.
I believe Dandanon also required that as well, but it's been that long since I read the rules...
Technoir has a variation on this : Any skill(verb) you fail at using during a mission becomes 'primed'. Then, between scenes when the characters are resting and treating their wounds, they get a chance at raising their primed verbs. Succeed and the verb's dice pool augments by 1, the verb losing its primed status. Failure means the verb remains primed and the player can still try raising them next downtime.