A.H.
M.,
I've flown several times with both my boys as babies and actually took my two year old and 1 month old on a flight to Iowa by myself. The 1 month old did no have a seat. If you're nursing that's probably the easiest way to travel. The key is to make sure you are nursing the baby when you take off and when you land. If she won't nurse then make sure she's sucking on something i.e. pacifier. This helps them keep their ears cleared. Other than that make sure you have an extra change of clothes, diapers, wipes and you'll be good. I wouldn't take your car seat with you. What I would do is ask if there is a row with an open seat or ask for the bulkhead. Some planes that are smaller will only allow lap babies in certain rows and they will tell you that.
Just relax and enjoy your trip. The baby will be fine. The flight attendants are usually really helpful and you will find that other passengers are quite gracious in offering assistance when they see the situation you are in.
Good luck!