In this article, we introduce a new general family of skewed distributions obtained through the use of a weighted skewed technique. This technique has the feature to unify two classical skewness techniques. Also, it is based on a clear stochastic representation involving a tuning weight function. General moments results are given. Subsequently, we focus our attention on a special case called asymmetric bimodal normal distribution. We investigate the maximum likelihood estimation of the parameters for this new distribution, with a complete numerical study. The developed model and method of inference are applied to some daily closing prices of some popular stocks.