<p>The utilization of copulas for modeling dependence has garnered significant attention in recent years. Conversely, the quest for multivariate copulas with desirable dependence properties remains a crucial area of research. When fitting regression models to longitudinal data, the multivariate Gaussian copula is frequently employed to accommodate the temporal dependence of repeated measurements. However, using a symmetric multivariate Gaussian copula may not be ideal in all scenarios, as it fails to capture non-exchangeable dependence or tail dependence if present in the data. Therefore, to ensure reliable inference, it is imperative to explore beyond the Gaussian dependence assumption. In this paper, we introduce the construction of a geometric skew-normal copula derived from the multivariate geometric skew-normal (MGSN) distribution proposed by [<CitationRef CitationID="CR24">24</CitationRef>] and [<CitationRef CitationID="CR25">25</CitationRef>], aimed at modeling the temporal dependence of non-Gaussian longitudinal data. Initially, we examine the dependence properties of the proposed multivariate copula and subsequently develop regression models for both continuous and discrete longitudinal data. Notably, the quantile function of this multivariate copula remains independent of the correlation matrix of its respective multivariate distribution, offering computational advantages in likelihood inference compared to the copulas derived from skew-elliptical distributions as proposed by Azzalini and others. Furthermore, composite likelihood inference becomes feasible for this multivariate copula, allowing for parameter estimation from ordered probit models with the same dependence structure as the geometric skew-normal distribution. We conduct extensive simulation studies to validate our proposed models and apply them to analyze the longitudinal dependence of two real-world datasets. Finally, we present our findings in terms of the improvements over regression models based on multivariate Gaussian copulas.</p>