Let ##(M,g)## be an ##n##-dimensional pseudo-Riemannian manifold of signature ##(n_+, n_-)## and define the Levi-Civita symbol via
$$\varepsilon_{i_1 \dots i_n} \, \theta^{i_1 \dots i_n} = n! \, \theta^{[1 \dots n]} =
\theta^1 \wedge \dots \wedge \theta^n$$
where ##\theta^1, \dots, \theta^n##...