See

Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. *Proc. Amer. Math. Soc.* **85** (1982), no. 2, 283–288.

for the complete answer to all of your questions. (The paper is referenced in other papers
and at least one of these has an application to singularity theory.)