Is this a problem?
No, not so much, because egg whites keep for a long time, you can very well put them in a bowl or a container, which you film and keep in the fridge.
As egg whites are only water and proteins, they keep very well, they don’t go mouldy and they keep all their qualities (well, they have almost no taste anyway).
Better still, for certain recipes such as macaroons, you will succeed more easily if you use “old” whites, the pastry chefs say “dry whites”, which contain less water than fresh whites.
At home, for example, I have, perhaps like you, a whites bowl in which I take or add whites as I produce or consume them. It is just filmed to protect them from the air, and the whites keep in it for 3 weeks easily.
And for longer?
If you are worried about keeping them in the fridge for a long time, you can also freeze them very well.
But it would be a mistake to freeze 6 whites at once for example, you will be forced to defrost all 6 at once as well, which is not very flexible.
That’s the trick, you have to freeze them individually by pouring them 1 by 1 into a silicone mould a bit like this one:
The shape doesn’t matter, it just needs to fit a whole white (about 50-60g).
If you have several whites in a bowl, because of their viscous aspect, they will not be easy to pour individually into the moulds.
So to make it easier you can give them a few whacks with a whisk, or even a quick blast with a mixer (2 or 3 seconds), this will make them more liquid and easier to pour/mould.
Then put in the freezer overnight to set, unmould into a freezer bag, and scoop in as you need to.
In summary: Egg whites keep well in the fridge, but they also freeze well, and in this case it’s best to freeze each white individually so you can then scoop out just what you need.