It can be anything in your cooling system, not only the thermostat. But that's the first thing to check and replace. It is designed to fail in open position which will result in always-on state for the cooling system. But your fan is always on, which shouldn't be the case also. If the cooling system is working well, even on a hot day the fan cuts in and out every few minutes. Could be the fan's thermostatic switch or coupling is kaput resulting in always-on state.
So it would seem that your car is over-cooling and that will result in one or more other problems IN THE LONG RUN. As I said in an earlier post, an engine running cool is not good, but a lot of laymen will think it is better to be cooler and our untrain trial and error mechanics will add to that fantasy.
See this site to understand how a thermostat work:
http://auto.howstuffworks.com/cooling-system8.htm
That's why I say a thermostat is a renewable item just like the spark plugs. They don't last forever. In fact if you've overheated the car before, some of the wax may have melted out and it will give a wrong result.