Classification of principal G-bundles

Question

Let $G$ be a group. We define the functor $$Bun_{G}^{*}:h(CW_{*}^{op})\rightarrow Sets_{*}$$ which takes a pointed CW complex and assigns to it the set of isomorphic classes of principal $G-$bundles over it. I have heard that we can construct a space $BG$ such that this functor can be representable (I have heard that this is a theorem by Milnor). However, I would be interested in reading a proof using the Brown representability theorem (the space $BG$ is constructed abstractly). Unfortynately, I cannot find such a result in the literature.