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.