  We introduce a new approach to the study of timelike minimal surfaces in the
Lorentz-Minkowski space through a split-complex representation formula for this
kind of surface. As applications, we solve the Bj\"orling problem for timelike
surfaces and obtain interesting examples and related results. Using the
Bj\"orling representation, we also obtain characterizations of minimal timelike
surfaces of revolution as well as of minimal ruled timelike surfaces in the
Lorentz-Minkowsi space.
