Classifying space of $GL_{n}(\mathbb{F})$? I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage to find something. Can you help me please? Also, do you know what's the cohomology ring of that group in the case where the field is finite or infinite?

I think that I can give an answer, at least for the very first part of my question. Assume that $GL_{n}(\mathbb{R})$ is the general linear over $\mathbb{R}$. It works almost in the same fashion for $\mathbb{C}$ as well. Then, someone can prove that the "natural map" $\rho:V_{n}{\mathbb{R}^{\infty}} \rightarrow G_{n}{\mathbb{R}^{\infty}}$ from the $n$-frames into the infinite dimesional real vector space to the $n$-dimensional Grassmanian of the same space is a $GL_{n}(\mathbb{R})$-bundle. Also the total space is contractible, so the Grassmanian is a model for $BGL_{n}(\mathbb{R})$. If we restrict our attention into the case where our total space is a Stiefel manifold of the $\textbf{orthonormal}$ $n$-frames, which I will denote by ${V^{O}_{n}} {\mathbb{R}^{\infty}}$ for simplicity, because of the Gram-Schmidt Theorem we end up by a deformation retract between the Stiefel manifold and $V_{n}{\mathbb{R}^{\infty}}$. So that gives an $O_{n}(\mathbb{R})$-bundle, $\tilde{\rho}: {V^{O}_{n}} {\mathbb{R}^{\infty}} \rightarrow  G_{n}{\mathbb{R}^{\infty}}$. Hence $BGL_{n}(\mathbb{R})=BO_{n}(\mathbb{R})$. 
Any comments, or corrections are really welcome!
Also any comment for cohomology of $GL_{n}(\mathbb{F})$ for any kind of field may be helpful!
