  Let $p$ be a prime and $\mathbb{F}_q$ be the finite field of order $q=p^m$.
In this paper, we study $\mathbb{F}_q\mathcal{R}$-skew cyclic codes where
$\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$ with $u^2=u$. To characterize
$\mathbb{F}_q\mathcal{R}$-skew cyclic codes, we first establish their algebraic
structure and then discuss the dual-containing properties by considering a
non-degenerate inner product. Further, we define a Gray map over
$\mathbb{F}_q\mathcal{R}$ and obtain their $\mathbb{F}_q$-Gray images. As an
application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray
images of dual containing $\mathbb{F}_q\mathcal{R}$-skew cyclic codes and
obtain many quantum codes with better parameters than the best-known codes
available in the literature.
