I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer is that $H^0$ and $H^n$ are $\mathbb Z$, and all others vanish, but that is only from trying to understand a proof (from Milnor and Stasheff's "Characteristic classes," where the Euler class is defined).
Any suggestions on how to show this?