# Cohomology of classifying space

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces:

1) $$H^\ast (BSU(2), \mathbb{Z})$$

2) $$H^\ast (BO(3), \mathbb{Z}/2)$$

3) $$H^\ast (O(3), \mathbb{Z}/2)$$

Thank you

1. Note that $$BSU(2) = BSp(1) = \mathbb{HP}^{\infty}$$, so $$H^*(BSU(2); \mathbb{Z}) \cong \mathbb{Z}[\alpha]$$ where $$\deg\alpha = 4$$.
2. In general, $$H^*(BO(n); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1, \dots, w_n]$$ where $$\deg w_i = i$$.
3. Note that $$O(3) \cong SO(3)\times\mathbb{Z}_2$$ and $$SO(3)$$ is diffeomorphic to $$\mathbb{RP}^3$$, so $$O(3)$$ is diffeomorphic to $$\mathbb{RP}^3\sqcup\mathbb{RP}^3$$. Therefore $$H^*(O(3); \mathbb{Z}_2) \cong \mathbb{Z}_2[\alpha, \beta]/(\alpha\beta, \alpha^4, \beta^4)$$ where $$\deg\alpha = \deg\beta = 1$$.