Summary: This article discusses skew generalized quasi-cyclic codes over any finite field \(\mathbb F\) with Galois automorphism \(\theta\). This is a generalization of both quasi-cyclic codes and skew polynomial codes. These codes have an added advantage over quasi-cyclic codes since their lengths do not have to be multiples of the index. After a brief description of the skew polynomial ring \(\mathbb F[x; \theta]\), we show that a skew generalized quasi-cyclic code \(C\) is a left submodule of \(R_1 \times R_2 \times \ldots \times R_\ell\), where \(R_i \triangleq \mathbb F[x;\theta]/(x^{m_i}-1)\), with \(|\langle\theta\rangle| = m\) and \(m\) divides \(m_i\) for all \(i \in \{1, \ldots, \ell\}\). This description provides a direct construction of many codes with best-known parameters over \(\mathrm{GF}(4)\). As a byproduct, some good asymmetric quantum codes detecting single bit-flip error can be derived from the constructed codes.