How long does eggwash keep?
To make a long answer short, if you are using it once a week, then don't bother storing the egg mixture. The egg wash is definitely not safe after a week even if you store it in an air type container in the refrigerator.

If however you are using just egg whites, you can lightly beat half of it and use that and freeze the other unbeaten half in the freezer. 

But to be honest... There are worse things to waste than half an egg a week. I think this is a case of not being worth the effort.