There is no documentation requirement for domestic travel in the US for children flying with a parent. I've flown dozens of times with no one saying boo.
How do I write a consent letter for my child to travel? List your child's name, birth date/place, and passport details. Provide the parent's/guardian's name, custody information, and passport details. Add contact information for the non-traveling parents/guardians. Detail the child's destination and dates of travel.
Typically, minors under the age of 18 do not have to present identification for domestic U.S. travel. Airlines will accept identification from the responsible adult on behalf of the minor(s). For international travel, minors under the age of 18 must present the same travel documents as the adult.
A letter of consent, preferably in English and notarized, from the other parent or signed by both custodial parents. The letter should say: "I acknowledge that my child is traveling outside the country with the name of the adult with my permission."
You should obtain a child travel consent letter granting you permission from the absent parent to leave the country with your child (having it notarized is not required, but highly recommended).
In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.