Quantifying dependence between extreme values is a central problem in many theoretical and applied studies. The main distinction is between asymptotically independent and asymptotically dependent extremes, with important theoretical examples of these general limiting classes being the extremal behaviour of a bivariate Normal distribution, for asymptotic independence, and of the bivariate t distribution, for asymptotic dependence. In this paper we study the tail dependence of skewed extensions of these two basic models, namely the bivariate skew-Normal and skew-t distributions. We show that both distributions belong to the same limiting class as the generating family, the skew-Normal being asymptotically independent and the skew-t being asymptotically dependent. However, within their respective limiting class, each provides a wider range of extremal dependence strength than the generating distribution. In addition, both the skew-Normal and the skew-t distributions allow different upper and lower tail dependence.