Binary regression models are known to often produce asymmetric posterior distributions, especially in situations where the number of observations is limited compared to the parameter dimension of the model. As a consequence, in these settings, the use of common symmetric approximations of the posterior distribution based on Gaussian densities can lead to biased inference. To overcome this problem, in this paper we propose the use of the skew-symmetric approximation method developed by Durante et al. (2023). Although this technique applies to various binary regression models, we specialize our results to the important case of logistic regression. In this framework, we show that the skew-symmetric approximation is not only theoretically justified in standard asymptotic regimes, but can also work well in practice when the number of parameters of the model is larger than the sample size.