Bijective correspondence between normal vector fields and n-forms

Question

Let $M$ be an $n$-dimensional smooth submanifold of $\mathbb{R}^{n+1}$. I'm having trouble convincing myself that, as one might intuit, there is a bijective correspondence between smooth normal vector fields on $M$ and elements of $\Omega^n(M)$. Let's say that the map is $$Y\mapsto\omega_Y$$ where $Y$ is a normal vector field and $(\omega_Y)_p(W_1,...,W_n) := $det$(Y(p),W_1,...,W_n)$. This is injective, and it would be surjective if we could find a nowhere-vanishing normal vector field $Y$, for then one could use the fact that $\omega_{fY}=f\omega_Y$ to get any element of $\Omega^n(M)$. But how does one get such a $Y$?