The algebraic structure of skew cyclic codes over $M_2$( ${\mathbb{F}}_2$), using the ${\mathbb{F}}_4$-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree $d(x)$ is a free code generated by $d(x)$. According to our findings, skew cyclic codes of odd and even lengths are cyclic and $2$-quasi-cyclic over $M_2({\mathbb{F}}_2)$, respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over $M_2({\mathbb{F}}_2)$ is considered in this paper.

Let $M_2({\mathbb{F}}_2)$ be the ring of matrices of order $2\times 2$ over finite field ${\mathbb{F}}_2$ and $\omega \in M_2({\mathbb{F}}_2)$ be a cubic primitive root of unity. For any even positive integer $t$, the weight distributions of the skew cyclic codes of length $3t$ with parity check polynomials $x^t-{\omega }^i,i=0,1,2$ and $(x^t-{\omega }^j)(x^t-{\omega }^k)$, $0\le j<k\le 2$ were determined.