  We construct a 3-dimensional cell complex that is the 3-skeleton for an
Eilenberg--MacLane classifying space for the symmetric group $\mathfrak{S}_n$.
Our complex starts with the presentation for $\mathfrak{S}_n$ with $n-1$
adjacent transpositions with squaring, commuting, and braid relations, and adds
seven classes of 3-cells that fill in certain 2-spheres bounded by these
relations. We use a rewriting system and a combinatorial method of K. Brown to
prove the correctness of our construction. Our main application is a
computation of the second cohomology of $\mathfrak{S}_n$ in certain twisted
coefficient modules; we use this computation in a companion paper to study
splitting of extensions related to braid groups. As another application, we
give a concrete description of the third homology of $\mathfrak{S}_n$ with
untwisted coefficients in $\mathbb{Z}$.
