A type of generalized Wronskian is used for the induction step in the proof of Roth's theorem in Diophantine approximation. There's a nice exposition in Schmidt's lecture note, for example, but undoubtedly it can be found in lots of other places.
Schmidt, Wolfgang M., Diophantine approximation, Lecture Notes in Mathematics. 785. Berlin-Heidelberg-New York: Springer-Verlag. 299 p.(1980). ZBL0421.10019.