How to use the skin of the sacrificial animal
It is not correct to give the skin to the butcher as his payment. If the skin is exchanged for something, which can be finished by using it, like money, then exchanging the skin, and using that thing is not permissible in the light of the Shariah. Obviously, money can only be used only when it is given to somebody and the desired thing is taken in lieu of that. Thus, selling the skin and using that money is impermissible in the light of the Shariah.
If the skin is sold, then giving the money received to those who are needy is necessary. In this situation, the ruling for this is same as that of Sadqa Fitr, i.e. making needy people absolute owner of the money. The amount gained by the sale of the skin can be given to poor children, widows and other needy people, who are not descendents of the Holy Prophet (Sallallahu alaihi wa sallam). Also, giving it to the needy students of the Islamic universities is not only correct and permissible, but better and Mustahab (commendable) and this will get double the reward, firstly, the reward of giving it to those who are needy, and secondly the reward of contributing to Islamic education and propagation of Islam.
As making needy people the absolute owner is necessary, the skin cannot be given for construction of mosque, as salary to Imam and/or Muedhhin, salary to teachers, for conducting Islamic lectures, as gifts to the speakers, for social service, for burial of unknown and unclaimed Muslim corpses because the concept of making someone the absolute owner is not found in these cases. For this reason, all these are impermissible.
If the skin is retained in its original state and used, then the Shariah makes an allowance for it. For e.g., making a waterskin, prayer mat, coat, cap or table cloth, etc., then there is no problem. In the same way, it is correct to exchange the skin for something, which remains the same by using it, like a book, etc. The skin cannot be given in lieu of any job/work, so giving it to the butcher as his wages is not permissible in the light of the Shariah. As mentioned in Durre Mukhtaar:
(Durre Mukhtaar, Vol. 5, Kitab Ul Azhiya, Pg. No. 231)
Translation: The skin of the slaughtered animal should be given away as Sadqa (charity) or it can be made into a bag, waterskin, meal cloth (Dastarkhwaan) or pail (to draw water from a well) or it can be exchanged for such a thing, which remains the same with using it. It is not permissible to exchange it with things, which are finished by using them, like money. If the meat or skin has been sold, then money should be given in charity.
Usually, the skin is sold and on selling it, it is Compulsory (Wajib) to give in charity the amount received from that sale. For this reason, it is necessary to give it to the needy people. If the amount is not spent where it ought to be spent even then the sacrifice is affected.
For the sake of the Holy Prophet (Sallallahu alaihi wa sallam), may Allah Most High guide us and all Muslims to sacrifice animal/s with sincerity and may He accept it as well.