One argument that works: Notice there is a fibration G→EG→BG so using the Leray-Serre spectral sequence given information about H^{*}(G;Q) and H^{*}(EG;Q) one can hopefully determine something about H^{*}(BG;Q). Since EG is contractible this gives us one piece of the puzzle and since G has the homotopy type of a finite CW-complex with some difficulty one can actually show H^{*}(G;Q) is an exterior algebra with generators of odd degree. Using our spectral sequence it then turns out H^{*}(BG;Q) is a polynomial algebra with generators of even degree and the desired result follows. This might be overkill, but it works at least!

Edit: I wrote the above for coefficients in Q, but the same argument should work for R. Essentially the important fact is that over Q one can ignore torsion and the same obviously holds for R.