I've been playing around with the LinkedIn API and Sinatra for about a week and wrote a tiny app. After you authenticate using your LinkedIn credentials, the app allows you to search your connections for those that have a skill you specify. The source code can be found on GitHub.
Since this was more about using the LinkedIn API gem with Sinatra the matching algorithm is very naive. One interesting thing I found though is the amatch gem which is a set of extensions that allow you to do approximate matching, searching and comparing of strings. In the app I used the gem's Jaro-Winkler metric to compare the search term.