I can't seem to find a reference on the web that gives the $\mathbb{Z}$ cohomology of the Grassmann manifold of real n-planes in infinite dimensional Euclidean space and also the Bockstein maps associated with the coefficient sequence


$$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z/2Z} \to 0.$$


The real question is which products of Stiefel-Whitney classes are really $\mathbb{Z}$ classes.



Integer cohomology of the Grassman manifold of n planes in $R^\infty$
I don't know if these have everything that you want, but see the following:


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


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


