You have a fairly small number of observations to make any kind of strong statement about the shape of the distribution. The reaction time literature in psychology has an extensive number of papers on this subject with techniques like QMPE and QMLE developed for developing the RT distribution. Typically it's examined to see if effects are in the tail or shift of the distribution. The short story there is that much under 20 observations doesn't yield very reliable results. You might look at Heathcote et al (2004) on QMPE.
It sounds like if you know what you do know, that you know what kind of data this is. That may lead to theory that can provide more support for modelling. The RT literature is specific to theory about how RT's are formed, why they're skewed, etc. (see linear ballistic accumulator model). Data of a different kind might be better served by different modelling techniques. It really depends on what the shape of the distribution is caused by what model it will best follow. It might be underlying Poisson, Weibull, Ex-Gaussian, etc.
In short, there is no answer to your question the way you've put it but there is some guidance that can be given on finding one.