My preferred measure for skew would be the difference between the implied volatilities corresponding to the strikes where the Black-Scholes $d_2 = 0$ and $d_1 = 0$.
The reason I prefer this to others you often come across is that when the smile is symmetric the $d_2=0$ implied volatility equals the $d_1 = 0$ implied volatility.
Furthermore, the implied volatility corresponding to $d_2=0$ is quite insensitive to correlation, whereas the $d_1 = 0$ implied vola is very sensitive to correlation.
Also, the $d_2=0$ implied volatility is related to the volatility swap strike and corresponds to the the mid-point of the Black-Scholes log-normal distribution. Hence in a sense you can regard the $d_2=0$ strike as the pivot point about which the skew "rotates" as correlation changes, and correlation of course impacts skewness.
But as far as I know there is no generally accepted measure for skew, I am just giving my opinion.
I suspect even that the $d_1 = 0$ implied volatility is the most correlation sensitive point on the skew, but I cannot prove that (yet).