Consider the Levi-Civita symbol $\varepsilon_{ijkl}$ where $i,j,k,l \in \{0,1,2,3\}$. Consider now the binary representation of $i,j,k$ and $l$, such that $i \to (i_0 \, i_1)$, $j \to (j_0 \, j_1)$, $k \to (k_0 \, k_1)$ and $l \to (l_0 \, l_1)$. Is there a closed form of the Levi-Civita symbol in the form of
$$ \varepsilon_{ijkl} \equiv \varepsilon_{(i_0 \, i_1) (j_0 \, j_1) (k_0 \, k_1) (l_0 \, l_1)} = \textrm{function of 2-dimensional Levi-Civita tensors ?} $$
I have been trying to find an answer by direct inspection without any success. Thanks!