I'll assume that you mean the Borel cohomology $H^*(EG\times_GM)$. If so, the answer is
$$ \mathbb{Z}[\![c_1,\dotsc,c_n]\!]\otimes\Lambda^*(a_1,\dotsc,a_n), $$
where $|c_k|=2k$ and $|a_k|=2k-1$. To see this, first note that you can replace everything by maximal compact subgroups without changing the homotopy type. This means that the relevant space is $EU(n)\times_{U(n)}U(n)^{\text{ad}}$, where $U(n)^{\text{ad}}$ refers to the space $U(n)$ with $U(n)$ acting on it by conjugation. This can be described in a different way as follows. For any complex vector bundle $V$ (with Hermitian inner product) over a space $X$, we can consider the fibre bundle $U(V)$ whose points are pairs $(x,g)$, where $x\in X$ and $g$ is a unitary automorphism of $V_x$. If we take $V$ to be the tautological bundle over $BU(n)$ then $U(V)$ is easily identified with $EU(n)\times_{U(n)}U(n)^{\text{ad}}$. For any $V$ and $X$ it is known that
$$ H^*(U(V)) = H^*(X) \otimes\Lambda^*(a_1,\dotsc,a_n), $$

and my claim above is a special case of this. The key ingredient in the proof is Miller's stable splitiing theorem for $U(n)$, in the equivariant form proved by Nitu Kitchloo; some additional details are in my paper "Common subbundles and intersections of divisors".