We derive the Kendall and Spearman rank correlation coefficients of the bivariate skew normal (SN) distribution. For a given correlation parameter, we provide conditions on the shape parameters, under which the SN is more dependent than the normal in terms of each of the two-rank correlations. We further show how our results can be used for rank-based estimation procedures of the correlation parameter and the equal shape parameter of the SN, whose consistency and asymptotic normality we establish.