Does anybody know a proof (or a reference to proof) of the following combinatorial question: Let $a,b,c,\ldots=1,\ldots,n$ enumerate elements of a set of size $n$, e.g. vertices of an $n-1$ symplex. Let $C_{ab}$ be a cocycle, i.e. an object satisfying $C_{ab}=-C_{ba}$ and $$C_{ab}+C_{bc}+C_{ca}=0.$$
This can be interpreted as the sum over edges of a triangle $abc$, where the weight that each edge is assigned is the cocycle. Then the following identity holds: $$\sum_a (-1)^{a+1} \prod_{b<c,\, b,c\not=a} C_{bc}=0.$$
This can be interpreted as a sum over $n-1$ simplices of an $n$ simplex, with what is summed over being the **product** of the cocycles for all edges of the corresponding $n-1$ symplex (with signs). Note that for $n=3$ the above identity is just the cocycle condition itself.

This identity was checked (with Mathematica) for some low $n$'s, so I am reasonably confident it holds for a general $n$. It is also so simple that it must be known (if true). This is why I am hoping that somebody will point out a reference. Yet another question is: Is there any cohomological interpretation of this identity?