In Classification of weak 3-groups, Qiaochu gave an excellent answer, in which, he mentioned cohomology classes $H^{4}(B^{2}\pi_{2};\pi_{3})$ can be viewed as quadratic refinement of Whitehead bracket (where $\pi_{2},\,\pi_{3}$ are (discrete) abelian groups and $B$ is delooping.) In this case, Whitehead bracket is a binary operation $\pi_{2}\times\pi_{2}\to\pi_{3}$.
He didn't point out any ref for this point there and I can't find any. So my first question is:
Q1: How to see above algebraic meaning of $H^{4}(B^{2}\pi_{2};\pi_{3})$?
I have found some ref on nLab Whitehead product, but it didn't say anything on the cohomology class (at least in an obvious way to me).
Besides, I want to ask a more general question. We know that algebraically, $H^{*}(K(G,1))$ can be modelled by group cocycles (with a suitable target). So my second question is
Q2: Is there an algebraic model for $H^{*}(K(G;n))$ for general non-negative integer $n$?
