God dag Mikkel!
This is a really fun word. Where I'm from (New Zealand), at least, skittish is not in the vernacular at all. Some people may have a sense of what it means, probably from reading books like you have said, but couldn't define it and hardly ever use it regularly. People are more likely to say 'scatty,' or that they have their 'head in the clouds,' for example. Thanks for reminding me of this word though, maybe I should use it and see how people react or understand.
All that being said, I'm not sure whether or not it is used in the UK, with its many dialects or the USA. I don't believe it is, but I might be wrong. Maybe someone can correct me there :D
In sum, if you are planning to use this word, then by all means try using it, but there is a good chance that people will not understand. I hope this is useful to you or at least interesting.