Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle $\xi$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.