  This paper considers a new alphabet set, which is a ring that we call
$\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes
over the ring are then investigated in details. We define a nondegenerate inner
product and provide a criteria to test for self-orthogonality. Results on the
algebraic structures lead us to characterize $\mathbb{F}_4R$-skew cyclic codes.
Interesting connections between the image of such codes under the Gray map to
linear cyclic and skew-cyclic codes over $\mathbb{F}_4$ are shown. These allow
us to learn about the relative dimension and distance profile of the resulting
codes. Our setup provides a natural connection to DNA codes where additional
biomolecular constraints must be incorporated into the design. We present a
characterization of $R$-skew cyclic codes which are reversible complement.
