I am trying to get a concrete handle on the isomorphism $H^4(K(\pi_2,2),U(1)) \simeq \{$quadratic forms $\pi_2 \to U(1) \}$. This is explained in Eilenberg and Maclane's http://www.jstor.org/stable/1969702 and its companion but I am having a hard time getting just what this 4-cocycle should assign to a 4-simplex in $K(\pi_2,2)$. I am primarily interested in understanding the map from the right to the left.

I have a guess at something which may be close, which is there is a canonical closed 2-form on $K(\pi_2,2)$ valued in $\pi_2$. Using the associated bilinear form of the given quadratic form, I can wedge this form with itself to obtain a closed 4-form valued in $U(1)$. I worry that instead of the ordinary square, I need to be doing some factoring, perhaps using the Pontryagin square instead.

Any help, especially with some intuition, would be much appreciated.