What is the wedge product of two vector fields

Question

I've seen the notion of a wedge product for differential forms, but I've recently come across the notation $\frac{\partial}{\partial x^i}\wedge\frac{\partial}{\partial x^j}$, where these are vector fields in $\mathbb{R}^n$. Apparently, this is supposed to be an object that takes two smooth functions and produces a smooth function. How exactly does this act though?