Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that:

$$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$

under the following assumptions:

1. $m$ can be any number we want, but I prefer $2n$.
2. $E$ should not contain the terms $\det{(A)},\det{(B)},\det{(C)},\det{(D)}$. that means that the matrix $\begin{pmatrix} \det{(A)} & \det{(B)} \\ \det{(C)} & \det{(D)} \end{pmatrix} $ is not the case...
3. There can't be any further assumptions on $A,B,C,D$

I've already checked the matrix $\begin{pmatrix} A & B \\ C & D \end{pmatrix} $ but it's not that...

Does anyone have an idea what $E$ can be?

Solution 1:

Note: This answer is wrong, as indicated by the comments below.

Let $M \oplus N$ denote the block-diagonal matrix $$ M \oplus N = \pmatrix{M&0\\0&N} $$ Then one solution with $m = 4n$ is $$ E = \pmatrix{A \oplus I & I \oplus B\\I \oplus C & D \oplus I} $$ where $I$ denotes the $n \times n$ identity matrix.