Mom passed away on August 2nd, her social security check was deposited in her checking account August 3rd. I reported her death to social security, will they take the check back?

August 3 payment was for July. She lived the entire month of July.
So no clawback.

SSA now is pretty timely on stopping payments within 2 weeks from date of death. So she should NOT get a check Sept 3. But should that happen, that Sept $ will get a clawback.

No, they should not take it back but it has happened so try not to spend it. If taken back its an error and you will need to call SS. Mom should receive nothing in Sept because she died in August. That check they will pull back.

Here is a good article from AARP:

https://www.aarp.org/retirement/social-security/questions-answers/returning-social-security-payment.html

Since August's check is for July's benefit, it does not seem like they will want it back.