options - Constructing an approximation of the S&P 500 volatility smile with publicly available data

Besides of the VIX there is another vol datum publicly available for the S&P 500: the SKEW.

Do you know a procedure with which one can extrapolate other implied vols of the S&P 500 smile with these data (or with other publicly available vol data)?

Addendum:
I created a follow up question here.

Answer

There is a known expansion of implied volatility in moments (I'll find the reference)

\begin{equation} \textrm{IV} = \textrm{vol} * (1 + \frac{\textrm{skew}}{6} * \textrm{LMM} + \frac{\textrm{kurt}}{24}*(\textrm{LMM}^2-1)) \end{equation}

where log-moneyness is

\begin{equation} \textrm{LMM} = \frac{\log{\frac{\textrm{strike}}{\textrm{forward}}}}{\textrm{vol} * \sqrt{T}}. \end{equation}

Use VIX for vol.

If I remember correctly SKEW index is $100-100*\textrm{skew}$, so $\textrm{skew} = \frac{100-\textrm{SKEW}}{100}$. Kurtosis is unknown, but you could try to use VVIX index and re-scale it in some way.

Or maybe another way would be to take the equation and regress for multipliers for VIX, VIX*SKEW, and VIX*VVIX using IV smile data.