No, because Zakah is only supposed to be given to people, not to objects or institutions.
Zakah can only be given to poor people, which means that they do not possess more than 3 oz of gold in addition to their personal basic needs. Hospitals and schools are not people so they cannot receive Zakah. However, the hospital or school may act as an agent to collect Zakah on behalf of others and distribute it to those who are poor. They may do this by distributing the money directly or by distributing medicine or school supplies needed by the eligible people, since it is permissible to distribute needed items as Zakah.
However, an agent can easily misuse Zakah money by claiming to act as an agent on behalf of the poor and keep the money in order to provide services to them. This is contrary to Islam since the poor are to receive the money or needed items directly and not be forced into getting services from one particular institution. This is unfair and locks people into having to only get services from the agent institution rather than being able to go to another hospital and use the Zakah money given over there instead. Building a well, hospital, or school with Zakah money is not allowed because those institutions will usually serve non-poor people as well. However, if it is possible to distribute the Zakah money to the poor and then ask them to give some of it back in order to build a hospital for the poor, that would be hypothetically allowed as long as there was absolutely no coercion or pressure for them to donate this money back. However, such a scenario is very unlikely since poor people would spend the Zakah money for their immediate needs instead of long-term needs.
Shaykh Mustafa Umar