The image of the Veronese mapping is a determinantal variety; an elementary proof

Let ${\nu }_d:{\mathbb{P}}^n⟶{\mathbb{P}}^N$ be the Veronese mapping of degree $d$ where $N=\left(\genfrac{}{}{0px}{}{n+d}{n}\right)-1$. By an elementary approach it is shown that ${\nu }_d$ is an isomorphism of ${\mathbb{P}}^n$ onto the projective variety defined by the $2$-minors of a matrix.