Certain matrices are more fundamental under certain interpretations, one could use the determinant as a matrix generator for example \[\begin{vmatrix} \textbf{i} & \textbf{j} \\ \textbf{-i} & \textbf{j} \end{vmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} =\sigma_x \\ \begin{vmatrix} \textbf{i} & \textbf{j} \\ \textbf{i} & \textbf{j} \end{vmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} =i\sigma_y \\ \begin{vmatrix} \textbf{i} & \textbf{j} \\ \textbf{j} & \textbf{i} \end{vmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} =\sigma_z\]
However higher dimensions can be expressed easily \[\begin{vmatrix} \textbf{i} & \textbf{j} \\ \textbf{j} & \textbf{k} \end{vmatrix} = \begin{bmatrix} 0 & 0 & 1\\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\]
And extra information added \[\begin{vmatrix} (\textbf{i}+\textbf{j}) & (\textbf{j}+\textbf{k}) \\ (\textbf{k}+\textbf{l}) & (\textbf{l}+\textbf{i}) \end{vmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1\\ 1 & 0 & -1 & 0\\ 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix}\]
Fermionic ladder operators \[\begin{vmatrix} \textbf{i} & 0 \\ 0 & \textbf{j} \end{vmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} =a \\ \begin{vmatrix} \textbf{j} & 0 \\ 0 & \textbf{i} \end{vmatrix} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} =a^\dagger\\ \begin{vmatrix} \textbf{i} & 0 \\ 0 & \textbf{i} \end{vmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} =aa^\dagger\\ \begin{vmatrix} \textbf{j} & 0 \\ 0 & \textbf{j} \end{vmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} =a^\dagger a\]
And we have that \(a-a^\dagger=\epsilon_{ij}\).