I would question the assertion that "normal" wine goes off after "about a day".
About the only thing that can go wrong with it is for it to oxidize, and let's face it, the amount of oxygen that's going to get through the small opening of the bottle neck, is pretty tiny.
Here's an idea: experiment. Get a mid-range, drinkable red wine. Pour yourself a small glass and drink it. Leave the bottle open at room temperature. The next day, pour yourself another small glass from the same bottle and taste it. I promise it won't make you ill. Is it as good as the first day? Revolting?
From experience, I believe you'll find the wine still enjoyable after a week -- even if it's not in the peak condition it was in when you opened it. It will last even better if you cork it.
(For white wine, of course, you'd keep it in the fridge, because you want it cold when you drink it. You could keep red in the fridge, but then you'd want to bring it back to room temperature for drinking)
Perhaps you're thinking that your palate isn't sophisticated enough to notice the spoiling. I say, if you can't sense it, you shouldn't worry about it. (I also say, if you can sense it, and it's not a super high-end wine, you're being too fussy :) )
For cooking, chefs tend to recommend that you shouldn't cook with a wine you wouldn't drink. I go along with that, although there may be economic reasons, depending on how much wine costs where you live.