Transformation from vector $x$ to its Vandermonde matrix $V(x)$

Question

Consider a vector $x \in \mathbb{R}^N$ and the Vandermonde matrix $V(x)$

$$V(x)= [1 \; x \; x^2 \cdots x^{N-1}]$$

where the power of each vector is to be considered element-wise. Can I write the transformation $V: \mathbb{R}^N \to \mathbb{R}^{N \times N} $ only in terms of matrix/vector multiplications?