No Weapon skill, but maybe ...

We still don't know how the weapon skill tree will look like, but we do know that any class can use any weapon, so my assumption is that every class might have some passives that relate to some/most/all weapons. And you'll be able to spec into the weapon you want to main.

But as for "level up with use" stuff, I'd personally be against it cause that just adds grind on top of grind to the game. And considering that you'll have to specialize in artisan skills I don't see why you'd be able to be a jack of all trades in weapon skills.