I don't think it's rude to ask, especially if it's a small item and the space is empty when the door is closing. I've been known to hide my purse behind my calves, too. Putting things in the overhead isn't a great solution when I have a window seat; even with the additional legroom of bulkhead, the middle and aisle passengers often have to get out to let me out, and I would much rather ask one fellow passenger to share a few inches of space under my chair than force two to unbuckle, climb out, and stand in the aisle while I access the overhead at the beginning and end of the flight.