I can't see or understand if it is true or not if all vector bundles on over an open set of $\mathbb{R}^n$ are trivial or not. Is there an easy way to see it?
The problem comes from the fact that we define what a local trivialization is for a bundle and we define an atlas which trivializies the bundle. Since every chart is diffeomorphic to an open set of the euclidean space, if all the open sets have trivial bundles it should say that a trivialization could be global. The idea is that a vector bundle of rank $r$ over some manifold, with a trivialing atlas $\{(U_{\alpha},\phi_\alpha,\chi_\alpha)\}$ is equivalent to some limit of the sheaf $C^\infty(U_\alpha,\mathbb{R}^r)$ over all the atlas. The point of the question is if such atlas can always be taken to be a maximal atlas.