In Wald's General Relativity derivation of the Komar formula for the mass of a static, asymptotically flat spacetime there is the following passage:
Using Killing's equation $\nabla_a\xi_b = \nabla_{[a}\xi_{b]}$, we may rewrite equation (11.2.4) as $$F = \frac{1}{2} \int_S N^{ab}\nabla_a\xi_b \mathrm{d}A = - \frac{1}{2} \int_S \boldsymbol{\epsilon}_{abcd}\nabla^c\xi^d, \tag{11.2.5}$$ where, in the first line [sic], $N^{ab} = 2 V^{-1} \xi^{[a}N^{b]}$ is the normal "bi-vector" to $S$, and, in the second line [sic] $\boldsymbol{\epsilon}_{abcd}$ is the volume element on spacetime associated with the spacetime metric and the integrand is viewed as two-form, $\boldsymbol{\alpha}$, to be integrated over the two-dimensional submanifold $S$ (see appendix B). (The orientation of $\boldsymbol{\epsilon}_{abcd}$ is chosen so that $\boldsymbol{\epsilon}_{abcd} = - 6 N_{[ab}\boldsymbol{\epsilon}_{cd]}$, where $\boldsymbol{\epsilon}_{cd}$ is the volume element on $S$.)
I don't understand this second step (the second "line", as Wald calls it) at all. Why exactly is this manipulation possible? How does the spacetime orientation appears? Why is this choice of orientation possible? I tried working a bit with differential forms (of which I have a very limited knowledge of) but couldn't even write the contraction on the "first line" in differential form language.