I was reading the paper "On the Multilinear Restriction and Kakeya conjectures" by Bennett, Carbery and Tao
For each $1\leq j\leq n$ let $U_j$ be a compact neighborhood of the origin in $\mathbb{R}^{n-1}$ and $\,\Sigma_j:U_j\to\mathbb{R}^n$ be a smooth parametrization of a $(n-1)$-dimensional submanifold. Define $$ Y_j(x):= \bigwedge_{k=1}^{n-1} \dfrac{\partial}{\partial x_k} \,\Sigma_j(x) $$
My question is:
What does $\wedge$ mean in this case? In the paper, they say one can view $Y_j$ as a vector field.
This is a long comment. Suppose that $n=3$, so what does it mean $\frac{\partial\Sigma}{\partial x_1}\wedge \frac{\partial\Sigma}{\partial x_2}$? This is a 2-form, but by $\textit{duality}$ you can see it as a vector, the cross product, which is normal to the surface. Just the same for higher dimensions, you have a $(n-1)$-form, that consists of $n$ components, by $\textit{duality}$ or $\textit{Hodge star operator}$, you can look at these components as a vector.
Good luck!