If G is an affine algebraic group, a G-bundle is the same as a monoidal functor from G-reps to coherent sheaves. The map one way is take associated bundle, the other involves reconstructing the structure sheaf of the G-bundle from the associated ones. Roughly, you think of the functions on the group as a ring ind-object in the category of representations, and take the corresponding ring object in quasi-coherent sheaves. The Spec of this sheaf of rings is the G-bundle.
For GL(n), you'e lucky, since its category has a simple description: it's (basically) the free monoidal category with a single generator of dimension n. Other groups are a little more complicated, but not much worse.