I was asked in an exam to prove the following expression, $$ \det(A) = \dfrac16 \epsilon_{ijk}\epsilon_{lmn}A_{il}A_{jm}A_{kn}$$ I know that it can be written in the following form, $$ \det(A) = \epsilon_{ijk}A_{i1}A_{j2}A_{k3}$$ I don't know how to proceed beyond it.
Will someone please help me, Thank you.
RHS: $$\frac 1 6 ϵ_{ijk} ϵ_{lmn} A_{il} A_{jm} A_{kn} \\ = \frac 1 6 ϵ_{ijk} [e_l, e_m, e_n] A_{il} A_{jm} A_{kn} \\ = \frac 1 6 ϵ_{ijk} [A_{il} e_l, A_{jm} e_m, A_{kn} e_n] \\ = \frac 1 6 ϵ_{ijk} [e_i A, e_j A, e_k A] \\ = \frac 1 6 ϵ_{ijk} ϵ_{ijk} \det(A) \\ = \frac 1 6 \, 6 \det(A) \\ = \det(A)$$ with $[a,b,c]$ being the vector product of $a,b,c$.