I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group:
$H^\*(BSO(n);\mathbb{Z}) = $ ? (for example for $n=10$) 


I also like to know $H^\*(BPSU(n);\mathbb{Z})$ (say for $n=3$), where $PSU(n)=SU(n)/Z\_n$
and $Z\_n$ is the center of $SU(n)$.


Thanks!



Group cohomology of orthogonal groups with integer coefficient
For a precise answer to your first question, see Theorem 1.5 of 


Brown, Edgar H., Jr. *The cohomology of BSOn and BOn with integer coefficients.* Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288. 


For your second question, note that there is an isomorphism $PSU(n)\cong PU(n)$ for each $n$, and that the cohomology $H^\ast(BPU(3);\mathbb{F}\_3)$ is worked out in 


Kono, Akira; Mimura, Mamoru; Shimada, Nobuo *Cohomology of classifying spaces of certain associative H-spaces.* J. Math. Kyoto Univ. 15 (1975), no. 3, 607–617.


