Consider a four-dimensional spacetime. Consider the following contraction between Levi-Civita symbols and tetrads $$\epsilon_{\alpha \beta i j}\,{\epsilon^{ij}}_k\, e^\alpha\!\wedge e^\beta\!\wedge e^0\!\wedge e^k,$$ what I have done to simplify this is the following: $$\epsilon_{\alpha \beta i j}\,{\epsilon^{ij}}_k\,e^\alpha\!\wedge e^\beta\!\wedge e^0\!\wedge e^k = 2!\,\epsilon_{mnk}\,e^0\!\wedge e^m\!\wedge e^n\!\wedge e^k = 2!\,3!\,e^0\!\wedge e^1\!\wedge e^2\!\wedge e^3$$
where the latin indices take values $\{1,2,3\}$ and Greek indices take values $\{0,1,2,3\}$.
Is this correct?