What happens if somebody wants to do this out of the
Extension: Sorry, I knew it badly! I didn't know it should be done on Dhul'hijjah.
Hajj is not to be done in Ramadan. it is to be done in Dhu al-Hijjah ( ذو الحجة) but i get your question :)
we don't do Hajj on any other month because that what our prophet told us, and also in the Quran: god said
Quran2:197 "Hajj is [during] well-known months, so whoever has made Hajj obligatory upon himself therein [by entering the state of ihram], there is [to be for him] no sexual relations and no disobedience and no disputing during Hajj. And whatever good you do - Allah knows it. And take provisions, but indeed, the best provision is fear of Allah . And fear Me, O you of understanding."
If you want to read more about Hajj you can go to this Wiki page https://en.wikipedia.org/wiki/Hajj
Peace brother ;)