How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

Question

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a direct proof of the homotopy lifting property, but I can't come out with a direct lifted homotopy. Obviously $\pi : S(r,n) \to G(r,n)$ is the one that maps $(e_1,\dots, e_r)$ in $e_1 \wedge \dots \wedge e_r$.