Levi-Civita symbol

Let $M$ be an oriented semi-Riemannian manifold of dimension $n$ and signature $(s, n - s)$ . Let $e^{μ}$ be an orthonormal basis of 1-forms on some chart. Define the Levi-Civita symbol for $1 \leq i_{j} \leq n$ by

ϵ_{i_{1} \dots i_{n}} = {\begin{cases} sign (i_{1}, \dots, i_{n}) & all i_{j} distinct \\ 0 & otherwise . \end{cases}