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Spatio-temporal data often exhibit skewed distributions, which pose challenges for accurate modeling. Skew Gaussian random fields are among the common approaches for analyzing such data, although some existing models suffer from computational complexity and identifiability issues. In this paper, a Bayesian framework is proposed for modeling skewed spatio-temporal data based on a flexible closed skew Gaussian random field, which possesses desirable properties such as identifiability and closure under marginalization and conditioning. By employing the Matérn correlation function, the proposed model provides adequate flexibility for capturing spatio-temporal dependence structures. Bayesian inference is performed using the Hamiltonian Monte Carlo algorithm, and a simulation study is conducted to compare its performance with conventional Markov Chain Monte Carlo methods. Finally, the performance of the proposed model was also evaluated using observed PM-10 air pollution data.