In the case of two vectors $u, v$ with dimensions $n\times 1$, their Hadamard product can be represented by the following matrix multiplication: $$\mathrm{diag}(u)v = \left[\begin{array}{ccc} u_{1} & & 0\\ & \ddots\\ 0 & & u_{n} \end{array}\right]\left[\begin{array}{c} v_{1}\\ \vdots\\ v_{n} \end{array}\right]=\left[\begin{array}{c} u_{1}v_{1}\\ \vdots\\ u_{n}v_{n} \end{array}\right]\equiv u\circ v$$
Is there a way to generalize this for Hadamard products of matrices?
Hadamard product (Schur product) of matrices is element-wise product (two matrices dimension have to be same).
Reference: Topics in matrix analysis.