The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.
To clarify a bit. The tautological bundle over the Grassmannian $\gamma_k\to Gr_k(\mathbb{R}^N)$ has a complement $\gamma^\bot$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$. It follows that $\gamma\oplus\gamma^\bot =Gr_k(\mathbb{R}^N) \times \mathbb{R}^N$. The connection in this paper is the connection induced from the trivial connection by projection.