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?



A simple (?) combinatorial question
Choose an arbitrary number $D\_n$ and define $D\_i = C\_{in}+D\_n$ for $i \neq n$. So $C\_{ij} = D\_i - D\_j$.


The identity we want to prove is
$$\sum\_{a=1}^n (-1)^{a+1} \prod\_{\begin{matrix} 1 \leq b < c \leq n \\ b,c \neq a\end{matrix}} (D\_b-D\_c) = 0.$$


The product is the Vandermonde determinant $\det (D\_j^k)\_{ 1 \leq j \leq n,\ j \neq a}^{0 \leq k \leq n-2}$. So the sum is the row expansion of 
$$
\det \begin{pmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & 1 & 1 & \cdots & 1 \\
D\_1 & D\_2 & D\_3 & \cdots & D\_n \\
D\_1^2 & D\_2^2 & D\_3^2 & \cdots & D\_n^2 \\
\vdots \\
D\_1^{n-2} & D\_2^{n-2} & D\_3^{n-2} & \cdots & D\_n^{n-2} \\
\end{pmatrix}.$$


Since this matrix has a duplicate row, its determinant is $0$.


