  In this paper, the investigation on the algebraic structure of the ring
$\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its
automorphism group, enable to study the algebraic structure of codes and their
dual over this ring. We explore the algebraic structure of skew-constacyclic
codes, by using a linear Gray map and we determine their generator polynomials.
Necessary and sufficient conditions for the existence of self-dual skew cyclic
and self-dual skew negacyclic codes over
$\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given.
