All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Allaah specified the recipients of Zakat and it is not permissible to give Zakat to people who are not eligible for it, Allaah Says (what means): {As-Sadaqaat (Zakat) are only for the poor, and for the needy, and for those employed to collect (the funds); and for attracting the hearts of those who have been inclined (towards Islam); and to free the captives; and for those in debt; and for Allaah's Cause (i.e. for those fighting in the holy wars), and for the wayfarer (a traveller who is cut off from everything); a duty imposed by Allaah. And Allaah is All-Knower, All-Wise.}[Quran 9:60]. Therefore, it is not permissible to give Zakat for building schools or the administrations of schools, or spend it on Da'wah projects, and the like; for more benefit, please refer to Fataawa 90534 and 81303.
As regards the practices which you mentioned in the question, if the money is paid to the poor while conditioning on him to take a small amount of it and pay the remaining to another person or institution that will spend it on building schools, then this Zakat which is given by the poor is not accepted because it is in reality paid in order to built schools and the like, and it was paid to the poor just as a way of ultimately paying it to schools. However, if there is no agreement between the poor and the giver of Zakat, and the poor donates some of the Zakat which is given to him out of his own will to someone who builds schools and the like, then this is permissible since the poor owned the Zakat and is permitted to dispose of it as he wishes.