Should I write a thank you letter if I did not get the job?

Should I write a follow up or a thank you letter if I did not get the job which I interviewed/applied for? I want to do this so I can find out why I did not get the job. I would like to be able to find out if at all possible as to why they did not hire me or what the other candidates had over me that enabled them to get the poition over me.