Does cooking wine go bad like regular wine?

Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)

See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."

So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.

I've successfully frozen leftover wine and then later used it in stews. I've never tried it with "cooking wine" though, because, I prefer to only cook with wine that I'd actually drink.

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.

Cooking wines have extra preservatives added so they will stay good for longer, though you should probably put it into the refrigerator.

Related / duplicate question: How long can I store cooking wine?

I know it's not what you asked, but another way to get around this problem is to cook with drinking wine, and then drink the leftover with the meal. Since some of the wine is already in the meal it will go together perfectly.

Wine's "badness" is somewhat subjective. I once bought a cheap bottle of wine at Trader Joe's and was surprised to find that it tasted sour when I opened it. I didn't drink it, but rather than wasting it, I cooked with it since it was somewhere between wine and vinegar and adding acidity is a very common way to improve a dish.