I'm reading Appendix C of "Characteristic Classes" by Milnor & Stasheff and something confuses me. When proving that the Pfaffian of the curvature form (of an oriented $2n$-plane bundle $\xi$ over a smooth manifold $M$, with a metric and a connection compatible with that metric) represents a multiple of the Euler class of $\xi$, the authors say: (here $K$ stands for curvature tensor)

The notation $Pf(K(\tilde{\gamma}))$ suggests that $\tilde{\gamma}$ has a connection on it, thus the base space must be a smooth manifold. On the other hand, $\tilde{\gamma}$ is required to be "universal", so it should be the tautological bundle over the oriented Grassmannian $\tilde{G}_{2n}(\mathbb{R}^{\infty})$, but then the base space is not a manifold, seemingly a contradiction. So what should this bundle $\tilde{\gamma}$ be?