I Want to Put Biological Father on Birth Certificate

My question involves paternity law for the State of: Virginia. I would like to put the last name of the my child's father on his birth certificate. I realize that I cannot list the father without him signing something but am I allowed to give my child the last name anyways? I do not think his father will be around but I still want my son to have his dad's last name anyways.

You can name your son Horatio Nelson Gates-Trump III if you wish :)