If we take the Pauli Matrices \[\sigma_1=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\\ \sigma_2=\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\\ \sigma_3=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\]
and perform the same row column additions as were made to \(\epsilon_{ij}\) to get \(\epsilon_{ijk}\) we have \[\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}=\lambda_6\\ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}=\lambda_4\\ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \to \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}=\lambda_1\\ \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{bmatrix}=\lambda_7\\ \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{bmatrix}=\lambda_5\\ \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \to \begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}=\lambda_2\\ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}=\lambda_3\\ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}=A\\ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \to \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}=B\\\]
Here \(\frac{1}{sqrt{3}}(A+B)=\lambda_8\) where \(\lambda_i\) are the Gell-Mann matrices.