I have $N^2$ equations which I can write as the following Hadamard product. Is there a way I can get rid of the Hadamard product and express this using usual matrix operations?
$\left[ \begin{matrix} 0 & a_{21} & \cdots & a_{n1} \\ a_{12} & 0 & \cdots & a_{n2} \\ \vdots & \vdots & \ddots & \vdots\\ a_{1n} & a_{2n} &\cdots & 0\\ \end{matrix} \right]\bigcirc \left[\begin{matrix} b_1 & b_2 & \cdots & b_n \\ b_1 & b_2 & \cdots & b_n \\ \vdots & \vdots & \ddots & \vdots\\ b_1 & b_2 & \cdots & b_n\\ \end{matrix} \right]=\left[ \begin{matrix} c_{11} & c_{21} & \cdots & c_{n1} \\ c_{12} & c_{22} & \cdots & a_{n2} \\ \vdots & \vdots & \ddots & \vdots\\ c_{1n} & c_{2n} &\cdots & c_{nn}\\ \end{matrix} \right]$
As stated in this thread given task is impossible in general. If you try to express it in terms of being $$ A \circ B = X \cdot B $$
for your case counter-example reduces to the following:
$$ A=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix},\, B=\begin{pmatrix} 1 & 1 \\ 1 & 1 \\ \end{pmatrix} $$ then $$ 2 = rank (A) = rank(A \circ B) = rank(X \cdot B) \le rank (B) = 1 $$
which is obviously incorrect. The same could be deduced for other cases of general A or B, satisfying your conditions.
I believe this still may be possible if you restrict A and B to meet rank, and maybe other conditions.