while using 'does' we use the form of the verb. The use of does in itself signifies singular so we don't need to use another singular verb.
eg. He knows… He doesn't know.
She eats… She doesn't eat.
While framing the question we say, " does he write well" ?
answer- yes he writes well. if we use does, the answer would be " yes he does write well."
hope it's clear.