Given a smooth rank $2^n$ real vector bundle $\pi:E\to M$ over a smooth manifold $M$. I want to determine sufficient conditions for $E$ to be isomorphic to an exterior bundle of some vector bundle. One way is to give conditions on the local trivialization charts of $E$. Suppose $E$ has an atlas $\left(U_i, \phi_i\right)$ of local trivializations such that:
$(i)$ For each $i$, $\pi^{-1}(U_i)$ is an algebra over $\mathbb{R}$, whose algebra operations agree on their intersections, and
$(ii)$ $\phi_i:\pi^{-1}(U_i)\to U_i\times \bigwedge \mathbb{R}^n $ is an algebra homomorphism .
Will these conditions suffice to show that $E$ is isomorphic to an exterior bundle of some bundle.