I want to change my maiden name to that of my deceased father..?

When I was born my mother was married (separated), but because she was married, I had to have her husbands last name instead of my biological father's last name. I recently got married and her husbands name appears on my birth and marriage certificate as my father. I want to honor MY father by taking on his last name. I don't quite know what to do, but I want to do something. It's a lie to look at another man's name where it doesn't belong. I collected SS benefits from my biological father so it is recognized somewhere that he is in fact my bio father. What to do?