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 $H^d(BO_n,Z)$, in term of $\delta(w_i)$ and $p_i$, where $w_i$ are the Stiefel-Whitney classes.

Here $\delta$ is the natural map from $H^d(BG,Z_2)$ to $H^{d+1}(BG,Z)$. So $\delta$ looks like the Bockstein homomorphism (of some sort).

My question is that can we regard $\delta$ as the Steenrod Square $Sq^1$ in the above Theoroms?