cohomology ring of a subspace of real projective spaces

Question

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology.

Now I want to find the cohomology ring $H^*(\mathbb{R}P^{n+2}\setminus \mathbb{R}P^{n};\mathbb{Z}_2)$ and $H^*(\mathbb{R}P^{n+2}\setminus \mathbb{R}P^{n};\mathbb{Z})$. How to get it? Any references? Thanks.

Note: I think $H^1(\mathbb{R}P^{n+2}\setminus \mathbb{R}P^{n};\mathbb{Z}_2)$ should be nontrivial. But $\mathbb{R}P^{n+2}\setminus \mathbb{R}P^{n}$ only has 3 nontrivial cells of dimension $n$, $n+1$, $n+2$. Why?