pseudovectors and covectors The cross product of two vectors, $\vec{c}=\vec{a}\times\vec{b}$, is sometimes characterized as a pseudovector or axial vector. On the other hand, if we look at the vectors $\vec{a}$ and $\vec{b}$ as components of 1-forms, then the elemens of the cross product can be seen as components of a 2-form, and $\vec{c}$ should be seen as the Hodge dual of this 2-form. This, to me, is the real nature of pseudovectors. My question is whether $\vec{c}$ can be seen as a covector, or a covariant 1-tensor. That is, if I write $a^i$ for the contravariat components of $\vec{a}$ and similar for $\vec{b}$, is there a tensor with components $T_{i,j,k}$ such that the covariant components of $\vec{c}$ are given by $c_k= T_{i,j,k}a^ib^j$?

This paper is relevant; it shows that the Levi-Civita symbol becomes a tensor when multiplied by the square root of the modulus of the determinant of the metric tensor.