Name That Story

Reading Margaret Atwood's Empson Lectures (Negotiating with the Dead: A Writer on Writing) I came across a passage in which Atwood describes a story in detail she recalls without being able to place.

I've lost the author's name, but I'm searching-- and it goes like this: A man living in a rooming-house spies on another roomer-- a dowdy young woman-- and learns that she is an alien, and that every evening when she comes home from work she takes off all her clothes, lies down on the floor, and attaches the top of her head to the head of a thin, flat, person-shaped skin. Then she empties herself into the skin, which fills up with her substance like a water balloon. The former empty skin is now the woman, and the newly emptied skin is rolled up and stored away. And so it goes, turn and turn about, until the voyeur can't refrain from meddling. While the woman is out he takes away the skin, and watches to see what will happen. The woman comes back and sees that the skin is missing. She can then do nothing but wait, in quiet despair. Shortly she bursts into flames and is burned to a crisp...

I recognized the story at once. Probably most people can guess who the author is from the description. But I'm wondering if this story is as well known as I think it is. (Needless to say, my interpretation of the story is quite a bit different from Atwood's.)