The Theorem 1.5 and 1.6 of

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

give a general answer for $H^d(BSO_n,Z)$ and $H^d(BO_n,Z)$, which are very complicated. I wonder what are $H^d(BSO_\infty,Z)$ and $H^d(BO_\infty,Z)$ for $d=1,2,3,4,5$?