Unless you are using a coffee mug as a qt tank, take a deep breath and a step back and let's analyze.
If you have fish in qt as a precaution before adding to display tank and suddenly the fish die its not necessarily due to one missed day of water changes.
I just had 4 fish die in QT in a 7 day period. Tank had 10 fish, 6 acclimated, ate and made it. 4 croaked, who the hell knows why, shipping stress, whatever.
If you flame died after 3 weeks, many things could have affected it.
In your water changes, were you careless with salinity and temperature?
Sometimes, newly acquired fish simply die for no obvious reason.
A small QT tank does not require a skimmer, just goo mechanical filtration and weekly water changes.
Again, new fish die suddenly at times.
Sick fish die suddenly as well, it depends of the disease or parasite and how advance the problem is, sometimes we notice disease after its too late.