If you have been notified by a company that they do not wish to take you on and you want to follow this up and perhaps find out the reasons they have decided not to hire you for the position they are recruiting for. You want to ask them what the reasons for, was it to do with skills, experience, look at the reasons they have put in your letter and ask to follow it up from there with more specific feedback.
Make sure your letter is factual and specific to the interviews and that it does not sound like you are begging for a second chance and asking them to reconsider their decision because this is not the reason for writing such a letter.
If you are looking for an idea of what you should be putting in your letter, please see the below example.
Follow Up Letter Example - Not Hired
Thank you for considering me for the role of NAME OF JOB ROLE
Thank you very much for allowing me the opportunity to attend an interview with your company for the position of NAME OF POSITION. I received your letter on DATE saying that you had found someone for the role that you were recruiting for and that they were closely matched to your skill requirements and company ideal.
I would appreciate it if you would be able to give me some feedback so that I am able to ensure that I can aim my responses differently in another interview and to ensure that I know why my skills were not closely enough a match to your organisation. I would be very grateful for this as it will help me with any further interviews I have and hopefully the continuation to my career.
Again, thank you for allowing me to get an insight to your company, I would like to take this opportunity to wish you well for the future and I thank you in advance for providing me with any feedback that you are able to.