Exterior derivative of a matrix?

Question

I was wondering if there is a name for a particular operation on a square matrix, I am curious as to what the following is called, or if it is used anywhere: Let $A=[a_{ij}]$ be a square $n\times n$ matrix, consider the matrix $$dA=[b_{ij}]:=[a_{ii}+a_{jj}+a_{ij}]$$ For those wondering about the weird notation, suppose I have an Abelian group $G$ and I look at a $1$-cochain $G\rightarrow \mathbb{Z}_2$, and say $G$ is nice, in that it is a vector space over $\mathbb{Z}_2$, and it comes with basis $e_1,\dots, e_n$. In general, a $1$-cochain is not determined by its value on "a basis", but for now, let's suppose we really only care about its value on $e_i$ and $e_i+e_j$. Say $\eta$ is this $1$-cochain, and that $\eta(e_i)=a_{ii}, \eta(e_i+e_j)=a_{ij}$, where $a_{ij}\in \mathbb{Z}_2$. Then, I wish to take it's co-boundary, and I am only interested in its value on pairs of basis elements, that is, I want $$d\eta(e_i,e_j)=\eta(e_i)+\eta(e_i+e_j)+\eta(e_j)$$ Or, if I were to write it as a matrix $$[d\eta]_{ij}=[\eta(e_i)+\eta(e_i+e_j)+\eta(e_j)]=[a_{ii}+a_{ij}+a_{jj}]=dA$$ In the end, I am interested in figuring out whether a given matrix is a co-boundary or not using this notion. For example, a $2$-cocycle whose class I am interested in determining is the $2$-cocycle $C:\mathbb{Z}_2^8\times \mathbb{Z}_2^8\rightarrow \mathbb{Z}_2$ given by the following matrix $$C=\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{bmatrix}$$ Is there a matrix $B$ so that $C=dB$, and if so, what is it?