I am trying to figure out whether an options structure is short or long skew without just having someone tell me the answer. I'd like to calculate the number myself. Assuming I am creating a risk reversal at strike 90, 110:
Does this make sense?
There are many definitions / concepts of skew. For a good overview of 'skew' you might like Mixon, What does implied volatility skew measure?
What you have done in your example is calculate the slope of the implied volatility smile/skew at two different strikes and then compared the slopes at these two different points. Mathematically speaking there's nothing wrong with this, but that's not the usual way to trade skew or even to quantify skew.
The most straightforward way to trade skew is to trade a risk reversal, i.e. long an OTM put and short an OTM call. You'll be long skew then because if the skew increases, i.e. OTM put IVs increase and OTM call IVs decrease, your structure will gain in value if you bought the risk reversal, all else equal.
The devil is in the 'all else equal'.
The cleanest way to trade skew is actually to trade skewness, which is the third (central) moment of the underlying. But for this you'll need a strip of options appropriately weighted. Skewness is not skew as traders understand it though.
To really trade the difference between two implied volatilities without the other 'noise' you have to trade forward start implied vols of certain strikes; this will give you almost pure exposure to forward start skew, but that may be a step too far at this moment.
