This is in some sense a specialization of the question integral or rational cohomology of real grassmannians. Let $G_3(\mathbb{R}^5)$ denote the real Grassmannian of (unoriented) $3$-planes in $\mathbb{R}^5$, which is a non-orientable closed manifold of dimension $6$. I would like to know the integral cohomology ring $$ H^*(G_3(\mathbb{R}^5);\mathbb{Z}). $$ Does anyone know of a reference where this is worked out, or how to go about doing so?