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I suggest you avoid using the VIX for implied vols. Why? One has to consider that the VIX is not simply solely dependant on the dynamics on the S&P 500 anymore because the VIX can be traded via options, etc. Thus many more parameters affect the trajectory of the VIX. The VIX has to equal the ATM option vol because this is where arbitrage assumption ...

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} = ...

Actually, closing options prices can be downloaded from the exchange, so the data necessary to get the skew is available. If for some reason you don't want to use those closing prices, it is possible to obtain a vol skew from VIX and SKEW. You would need to fit the parameters of a stochastic volatility model (such as Heston's) to the VOL and SKEW data. ...