When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in
- Kameko, Masaki; Yagita, Nobuaki, The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups ${\rm PU}(p)$ and exceptional Lie groups, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2265–2284, doi:10.1090/S0002-9947-07-04425-X
and in
- Vistoli, Angelo, On the cohomology and the Chow ring of the classifying space of ${\rm PGL}_p$, J. Reine Angew. Math. 610 (2007), 181–227, doi:10.1515/CRELLE.2007.071, arXiv:math/0505052.
For $n = 3$, the second paper contains a computation of the multiplicative structure.
