De Rham cohomology ring of the Stiefel manifold in low dimensions

Question

Let $V_k(\textbf{R}^n)$ be the Stiefel manifold : the $k$-frames in the $n$ dimensional real space. I'm trying to understand the de Rham cohomology ring for $k=2$ or $k=n-1$. I had good ideas for $n-1$ since it's the real projective space, but I'm having trouble for $k=2$. I don't want to use something too fancy (like the Serre spectral sequence), but I'm trying to use the sphere fibrations or the principal fibrations with the Grassmannian. How could one do it ?