Cohomology of BO(n) and BU(n) with coefficients in arbitrary ring

Question

We know that $$H^*(BU(n);\mathbb{Z}) \cong \mathbb{Z}[c_1,c_2, \ldots,c_n]$$ where $c_i$ is the $i$th Chern class and $$H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}_2[w_1, w_2, \ldots, w_n]$$ where $w_i$ is the $i$th Stiefel-Whitney class. My question is: what are $H^*(BU(n);R)$ and $H^*(BO(n);R)$ when $R$ is some commutative ring? What can we say about the two?