On US carriers, there isn't a prohibition against installing a carseat rear-facing. However, due to the seat pitch in coach, a lot of infant carseats will not fit between the rows installed that way; they may be too large. If so, installing the seat forward-facing is still safer than not using one at all.
With most models, the base will not be usable in an aircraft. The reason for this is that an aircraft seatbelt buckles in the center, rather than on the side. Depending on the way the base and seat fit together, the buckle in the center may prevent the seat from properly snapping into the base. I'd say you can try to fit a deck of cards between the seat and the base in the center; if you can't do that, then the base won't work. (A deck of cards being about the same size as the seatbelt buckle.) If the base won't work, you just install the car seat only, without using the base.
*All* currently-sold US carseat models must be designed to be safely installed using a lap-belt only; just follow the seat's directions for lap-belt installation. (If you have lost the instructions, call the mfr. They will send a new set for free.)