The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.

However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is used for multiplication in the exterior algebra.

E.g. I saw the following: let's prove the equivalence of linear dependence of three vectors $u_1,u_2,u_3$ in the space and linear dependence of their pairwise products $u_i\wedge u_j$:

take

$$ u_k\wedge \sum_{i<j} a_{ij} u_i\wedge u_j = 0 $$

and derive from this that $a_{ij}=0$, as $u_1\wedge u_2\wedge u_3=0$ means linear dependence of the $u_i$'s. (which would be fine if we talked about exterior product, but totally wrong for vector product, as it isn't even associative to begin with...)

Are there any arguments for using $\wedge$ for cross product?