In this article wiki appears $e_1,e_2,e_3$ in the determinant representing a cross product but it is never defined anywhere - is this some special cross product where $e_i$ can be every thing - an element from an alphabet, a complex number or even a set - maybe the alphabet itself?
I never seen this notation and I don't know what to look for to understand it.
In the identity $$\phantom{(\ast)} \qquad {\bf x} \times {\bf y} = \det \pmatrix{{\bf e}_1 & {\bf e}_2 & {\bf e}_3 \\ x^1 & x^2 & x^3 \\ y^1 & y^2 & y^3} = \sum_{i, j, k} \epsilon_{ijk} {\bf e}_i x^j y^k \qquad (\ast)$$ and in the context of Euclidean geometry more generally, ${\bf e}_i$ usually denotes the $i$th standard basis vector, so that (in $\Bbb R^3$) $$ {\bf e}_1 := (1, 0, 0)^{\top}, \qquad {\bf e}_2 := (0, 1, 0)^{\top}, \qquad {\bf e}_3 := (0, 0, 1)^{\top} . $$ Owing to the invariance of the cross product---that $A({\bf x} \times {\bf y}) = (A {\bf x}) \times (A {\bf y})$ for orthogonal $A$ with $\det A = 1$---we can replace $({\bf e}_i)$ in $(\ast)$ with any oriented, orthonormal basis, and the identities will still hold.