DD has been on 40 flights in her 18 months (believe it or not!). She is a very good flier, thank goodness! We live in Northern Canada, and the rules here are that you have to buy them a seat is they are over 2 years old. You can optionally buy them a seat if they are under 2. If you do this, then you can choose to have them strapped in their seat, or you must have them sitting on your lap for take off, landing and turbulence. DD has had her own seat, and not, and both have their pros. If you buy a seat, without using a carseat, then you have the extra room (assuming the plane is full - many planes will try and get you the space if they have it even if you haven't bought a seat, which is always a bonus!) Now that DD is 18 months old, I find it easier without one, since she will not sit in her seat for that long without getting fidgity/fussy (we have to do 2-3 different planes, flying over a full day, but nothing more than 3 hours at once). Plus, if she won't sit in the seat, then the seat just gets in the way! Even I have a hard time seating in my seat for more than 2 hours or so!
BTW, in Canada, any child who has a car seat must sit next to the window so that they don't block another person in the event of an emergency (just noted about her sitting between you and DH).
HTH!