What to do when a prospective PhD supervisor doesn't follow up after an initial positive response?
The professor encouraged you to apply --- so that's what he expects you to do next. It is nothing wrong in your attempt to establish a more substantial connection with him and to promote yourself in his eyes, it increases your chances to get selected. However, imagine that at the same time there are probably 100+ other candidates, trying to do exactly the same --- sending him several smart research questions, involving him in some discussion, and expecting to chat over Skype with him in a few days. How would you feel in his shoes? 

Of course, professors want to be available for their potential PhD students, TAs and colleagues, and of course, they want to participate in the selection process. But many of them do not want to carry out all the communication themselves --- that's what secretariat, admissions, and HRs are for! I assume there is a formal route to submit your application, and I suggest exactly what your professor suggested --- prepare your application and submit as instructed. 

If in a meantime you receive a message from him --- go forth and suggest a Skype meeting. If not --- assume that the professor is busy, and submit your application before the deadline anyway.