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I'm interested in the cohomology of the classifying space $BO(n)$ of real vector bundles of fiber dimension $n$ with coefficients in the integers, twisted by the $O(1)$ bundle defined by the class $w_1 \in H^1(BO(n),\mathbb{Z}/2)$. Let me denote these coefficients $\mathbb{Z}^{w_1}$.

Is it true $H^\star (BO(n),\mathbb{Z}^{w_1})= H^\star(BSO(n),\mathbb{Z})$?

Many thanks.

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