  We give an ADHM description of the Quot scheme of points ${\rm
Quot}_{\mathbb{C}^{n}}(c,r),$ of length $c$ and rank $r$ on affine spaces
$\mathbb{C}^{n}$ which naturally extends both Baranovsky's representation of
the punctual Quot scheme on a smooth surface and the Hilbert scheme of points
on affine spaces $\mathbb{C}^{n},$ described by the first author and M. Jardim.
Using results on the variety of commuting matrices, and combining them with our
construction, we prove new properties concerning irreducibility and reducedness
of ${\rm Quot}_{\mathbb{C}^{n}}(c,r)$ and its punctual version ${\rm
Quot}_{\mathbb{Y}}^{[p]}(c,r),$ where $p$ is a fixed point on a smooth affine
variety $\mathbb{Y}.$ In this last case we also study a connectedness result,
for some special cases of higher $r$ and $c.$
