Recognizing a $4\times4$ matrix Suppose I have
$$
\Sigma=\begin{pmatrix}\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}\end{pmatrix},\quad S=\begin{pmatrix}s_{11} & s_{12} \\ s_{21} & s_{22}\end{pmatrix}
$$
both of which are actually covariance matrices of two $2\times 1$ random vectors. So, in particular, $\Sigma$ and $S$ are symmetric. Now I have a $4\times 4$ matrix:
$$
A=\begin{pmatrix}
\sigma_{11}s_{11} & \sigma_{11}s_{12} & \sigma_{12}s_{11} & \sigma_{12}s_{12} \\
\sigma_{12}s_{11} & \sigma_{12}s_{12} & \sigma_{22}s_{11} & \sigma_{22}s_{12} \\
\sigma_{11}s_{12} & \sigma_{11}s_{22} & \sigma_{12}s_{12} & \sigma_{12}s_{22} \\
\sigma_{12}s_{12} & \sigma_{12}s_{22} & \sigma_{22}s_{12} & \sigma_{22}s_{22}
\end{pmatrix}
$$
I would like to write $A$ as a function of $\Sigma$ and $S$ (without referring to $\sigma$'s and $s$') using matrix operations ($\otimes$, multiplication, inversion, addition, and maybe something else I cannot think of). How do I do that please?

I might be wrong, but one way to do it may be through combining Kronecker product and Hadamard product:
$$ \biggl(\begin {bmatrix} 1 & 1 \end {bmatrix} \otimes S \otimes \begin{bmatrix} 1 \\ 1 \end {bmatrix}\biggr) \circ \biggl(\begin{bmatrix} 1 \\ 1 \end {bmatrix}\ \otimes \Sigma \otimes \begin {bmatrix} 1 & 1 \end {bmatrix}\biggr) $$ 
