Azzalini's skew-normal distribution is obtained through a conditional reduction of a multivariate normal distribution parameterized with a correlation matrix. It seems natural that when the parameterization of that multivariate normal distribution is complexified, more flexible skew-normal distributions could be obtained. In this note this specification strategy, previously explored by Azzalini [Scand. J. Stat. 33 (2006) 561-574] among many other authors, is formally analyzed through an identification analysis. © Brazilian Statistical Association, 2012.