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?


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