The short answer is no. Not right now.
Currently, moves made by other players must be deduced and determined from the outcome of the board. As well, not everyone on this site understands shorthand notation, and may find it confusing.
Of course, in the grand scheme of things, order and turn history for all players, as well as the option to choose between shorthand and full sentence are certainly possible, and would be welcome. The problem being is that, current adjudicator problems are far more pressing than this, more superfluous, issue. As well, keep in mind, this is a voluntary project, currently developed by only two people that I know of. Be patient, and since this is open source, feel free to help develop if you have the skills and time.
Until then, memorize. =D