# Terminology : definition of skew for a volatility smile

Is there a generally accepted definition of skew for volatility smiles? Is skew always defined as $$\frac{\partial{\widehat{\sigma}(K,T)}}{\partial{K}},$$ where $$\hat{\sigma}:[0;+\infty)\times[0;+\infty)\to[0;+\infty)$$ is the implied volatility surface ?

Is it typical to work with ATMF skew and not look at other strikes/moneyness points?