As Prudentia states, York is a fascinating city but it's quite a long drive from Skipton and you really need at least a full day to get the best from it.
I think the contributor who suggested Bradford probably had his tongue firmly in his cheek but I love the place because it's home to the wonderful National Media Museum:
http://www.nationalmediamuseum.org.uk/
However, if you want to visit a 'touristy' (but still lovely) place, within easy reach of Skipton, my choice would be Howarth:
http://www.bbc.co.uk/bradford/sense_of_place/h aworth_1.shtml
(Note that that's the first of two pages. Click the 'More' button, at the foot of the page, to read everything).
Chris