Cross product in curvilinear coordinates in 7D

Question

I am wondering if there exists a formula for cross product of two vectors in seven dimensions in general curvilinear coordinates. According to Wikipedia cross product only makes sense in 3D and 7D. For two vectors ($\vec{a}$ and $\vec{b}$) in three dimensions the formula is:
$\vec{a}\times\vec{b}=\sqrt{\det{g}}\:\varepsilon_{ijk}\:a^i\:b^j\:\vec{e}^k$
where:

1. $\det{g}$ is determinant of metric tensor
2. $\varepsilon_{ijk}$ is Levi-Civita symbol
3. $a^i$ is i-th contravariant component of vector $\vec{a}$
4. $b^j$ is j-th contravariant component of vector $\vec{b}$
5. $\vec{e}^k$ is k-th contravariant basis vector

Is there a similar formula for 7D?