Is there a way I can find out why I didn't get a certain academic position?
You haven't said how many applications you've submitted or whether you've had any interviews.  It's possible that you're just being unrealistic about how many applications it will take to obtain a position.  

In the US, employers are reluctant to say anything about why an applicant was rejected because of the possibility that this information will be used against the employer in a lawsuit.  It's likely that the faculty involved in the hiring decision will be under instructions to not say anything about why applicants weren't chosen.  

Furthermore, even if you could get one of them to talk off the record, the likely reason would be something like "other applicants were slightly stronger or were better fits for our needs."  The reason for this is that the academic job market in most fields is extremely competitive for job seekers, and thus most positions attract many (dozens or even hundreds) of very well qualified applicants.  You should not expect that your application will have better than a 1 in 20 chance of being selected for an interview.  

If you're making to the interview stage but haven't gotten any offers, then it's possible that your performance in the in-person interviews has hurt your chances of getting the job.  However, you shouldn't conclude that until you've had several interviews with no offers.