How does an estate bank account get established?

You want to know how to establish an estate bank account. The first question is who was authorized by your mother to handle her affairs after her death. The usual way to accomplish that is by naming an executor in a will. Did your mother leave a will? If you, were you named as her executor?

If you are her executor, then you need to learn what the law of your mother's state requires of executors. Often, state law requires that a court petition be filed (usually called a probate proceeding) and that you obtain court confirmation as executor before you can be authorized to open an estate bank account If you are not her executor, you need to contact whoever was named as her executor. And if your mother did not leave a will, you will likely need to filed what's called an "intestate" (legalese for dying without a will) probate proceeding

Even if your mother's state does not require that a probate proceeding be filed for her estate, a bank may insist on some type of legal authorization before allowing you to open an estate bank account.

I'd start with by checking with banks, beginning with whatever bank your mother used, to see what they want to open an estate account.