I am very sorry for your loss.
OK, if there was a funeral parlor involved in any arrangements for her after she passed, then OFTEN that entity will advise SSA of the passing. However, you want to be sure, so I'd call them to confirm. If they did not, or you didn't have a funeral director, then you can call SSA and advise them. You can call your local office or the national number at(###) ###-#### They will typically want you to send an original copy (certified) of the death certificate, to make sure it is not a hoax.
In the meantime, any benefit that is sent to her before the paperwork is processed, in January or after, is NOT rightfully sent or received, so do not spend it. The SSA will be demanding it back.
You may also want to look into whether you can get survivor benefits, both monthly and the $255 death benefit, so consider making a trip to your local SSA office about that, bringing your photo ID, marriage certificate, your birth certificate, and her death certificate. In fact, you can probably do all of the above in one shot by going down there with documents in hand.