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 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.