I like the red onions in my toss salads. Seldom do i use a whole one, because they are usually so large and i want to know how can i keep the already cut onion from going bad. I have tried just about everything, zip lock bags, wrapping in foil or plastic wrap, brown paper bags etc. But for some reason and sometimes the onion becomes slimy on the cut part, at which point i have to cut away a portion of the onion before i can use it.
Take a decent sized cling film, put the onion with the cut part down in the centre. Close the cling film over the round part and twist all the air out of the wrap.
The onion will stay good in the fridge for some days.
BaffedCook's answer responds to your question more directly. I'm suggesting a roundabout approach.
Slice the rest of the onion and pickle it. Try your salads with pickled onions instead of raw red onions. Try your other foods with the pickled onions. Salty foods tend to go well with acidic foods.
Pickling is an age-old technique used for preserving foods that still exists to this day because it adds novelty to foods. Anything pickled should last for at least a month more than its normal shelf life.