I don't advise a hit on the butt because she's already out of control and hitting just instills fear. For the temper tantrums in the mall, I recommend

1. Talk with her before you go, about what you expect of her. No tantrums, no begging for things.
2. If she's good you'll offer her a reward and tell her this before hand. Yes it's bribery lol
3. Brings snacks with you in case she's hungry/thirsty.
4. Don't go when she's tired or hungry.
5. If she has a meltdown take her out of the mall immediately without getting angry. Just pick her up and say we're going.

At home you can do time out as Yoda mentioned. But with time out you need to be prepared to enforce it and be consistent. Ask her about school and if there were any problems. Often my girls will tell me about bad behavior of other kids and then we'll discuss what would have be better to do or say. Sometimes if my 8 yr old has a tantrum I'll just let her get it out and wait until she's calm - the we talk about it. Usually by then, she's ready to acknowledge her behavior and say sorry. This is a normal stage and my girls have had their share of tantrums. Good luck