The three-dimensional cross product can be reconstructed from two more general ideas: determinants and Euclidean duality.

For fixed vectors $\mathbf{u}$ and $\mathbf{v}$, the map

$$\mathbf{w}\mapsto \det(\mathbf{u},\mathbf{v},\mathbf{w})$$

is a linear functional of $\mathbf{w}$. Euclidean duality therefore associates it with a unique vector $\mathbf{p}$ such that

$$\mathbf{p}\cdot\mathbf{w}=\det(\mathbf{u},\mathbf{v},\mathbf{w})$$

for every $\mathbf{w}$. That vector is precisely

$$\mathbf{p}=\mathbf{u}\times\mathbf{v}$$

This explains at once why the cross product is perpendicular to both inputs, why its magnitude equals the oriented parallelogram area, and why swapping the inputs reverses its direction.

From oriented area to determinants

The determinant connection in three dimensions

Deriving the cross product from linear transformations