My no-frost fridge fan stopped working and it caused frost buildup at the back of the freezer and eventually the fridge stopped chilling the lower compartment. I assumed that the fan was broken but it turned out that there was no power at the fan socket and the fan was fine. I can't reach the back of the fan socket but I can connect the fan directly to the mains power.
Would it cause any problems? Is it supposed to work all the time (except when the door is open) or is should it only work when the compressor is working?