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 an answer to your questions. (As for applications: the paper is referenced in other papers
and at least one of these uses it to obtain a result in singularity theory.)

If I remember correctly, the computation is quite elaborate, even to state.


**Added:** the following also does the computations.

Feshbach, Mark,
The integral cohomology rings of the classifying spaces of $\text{O}(n)$ and $\text{SO}(n)$. 
*Indiana Univ. Math. J.* **32** (1983), no. 4, 511–516.