Simply because there is no reason to write "écrivez" here. If "vous" was the subject and "écrire" was the verb then in a sentence like: "vous écrivez", yes, you would have to write "écrivez" but here the subject is "je" and the verb is "aimer" then "écrire" or "(vous) écrire" is the thing you like (kind of a complement of the verb), so it has to be in the infinitive form. For exemple: "j'aime faire du vélo", "vous aimez peindre", etc. I have no idea of how to explain that correctly and someone else might do it much better than me, but I hope it helps anyway.

J'aime vous écrire = j'aime écrire à vous (I like to write to you) J'aime lui écrire = j'ai écrire à lui (I like to write to him) We don't write à vous, à lui like this at the end, we prefer to write it like in the sentences on the left of = . à moi = me ou m' à toi = te ou t' à lui = lui à elle = lui à nous = nous à vous = vous à eux = leur