The dictionary form is "맵다." This is called a "ㅂ irregular" in which the ㅂ gets replaced by 우 when a vowel is added to the stem (맵). So you start with 맵다 from which you get the stem "맵." You conjugate it to "맵은" like you said, but since a vowel is added it changes to "매운."
The base form is 맵다, which is irregular, and ㅂ changes to 우 when conjugated.
All the above responses + You add ㄴ at the end of a verb stem if it ends in a vowel, and 은 if it ends in a consonant. Therefore, 매우 + ㄴ = 매운 (spicy). 작 (verb stem meaning "short") end in a consonant, so 작 + 은 = 작은 (short)
Hope this helps (: