The paper “[Existence of universal connections](https://doi.org/10.2307/2372896)” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmannian 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 Grassmannian 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
$\gamma_k\to \operatorname{Gr}_k(\mathbb{R}^N)$ has a complement $\gamma_k^\perp$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$.  It follows that
$\gamma_k\oplus\gamma_k^\perp =\operatorname{Gr}_k(\mathbb{R}^N)  \times \mathbb{R}^N$.  The connection in this paper is the connection induced from the trivial connection by projection.