It is fortunate that you learned about the identity theft sooner rather than later. I doubt that the SSA will do anything about this. Please take a moment to review my Legal Guide on steps to take when you learn you are the victim of identity theft. The link is below.
Once you have completed several of the steps and have a police report of identity theft, you will find the bank much more cooperative to close the false account. With the police report and FTC report, they are required to treat you as the victim, not as someone coming off the streets claiming to have the same SSN as another account holder. As you know, even Social Security cards can be phony, so they treat everyone with some degree of suspicion, and would not close the account until they have a police report.
I have been representing consumers since 1997, including many cases of identity theft. Please contact me through my Avvo profie, if you need further assistance.
Report this (please see the link below), and make sure your credit is "frozen" so this person and no one else can open any accounts using your personal information.
Please also see the CA ID theft laws linked below.
So far you have no damages, so there's no reason to sue anyone, just stop this from going any further.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.