In this work we generalize the Weierstrass representation formula for surfaces in the Minkowski space. The Weierstrass parametrization is a local conformal parametrization of a surface which represents every minimal surface in \(\mathbb{E}^3\) by a pair of complex functions \((f,g)\), where \(f\) is holomorfic and \(g\) a meromorfic function. In the first part we introduce the basic terms and results on surfaces in the Minkowski space with focus on lightlike surfaces and dual functions, which we will use as the tool for representing lightlike surfaces. We analyse the parametrizations for spacelike and timelike surfaces which are already known and show that McNertney’s formula represents all regular spacelike surfaces when the functions \((f,g)\)) are not necessarily holomorphic. We find the Weierstrass representation for lightlike surfaces using a pair \((f,g)\) of dual functions, where \(f\) is holomorphic and \(g\) a meromorphic function. Every lightlike surface in \(\mathbb{M}^3\) is minimal. Using the link with dual numbers, we find the lightlike catenoid and helicoid and we construct the entire associated family and the adjoint surface of a lightlike surface. We show a 1-1 correspondence between the conformal parametrizations of a maximal/minimal surface and the conformal parametrizations of a unit sphere. In the second part we generalize every Weierstrass parametrization for 2-surfaces in \(\mathbb{M}^4\) analogously as in the Euclidean space. For spacelike surfaces this is already done by using complex functions, for timelike surfaces we use complex functions \((q,r)\) of two real variables and real functions \((f,g)\) and for lightlike surfaces we use dual functions \((\rho, f,g)\). For each representation we find the class of functions which represent maximal/minimal surfaces. We generalize the parametrization for spacelike surfaces for 2-surfaces in \(\mathbb{M}^n\). In general relativity we find a local parametrization of the so-called trapped surfaces in the Schwarzschild space, the simplest model of spacetime containing a black hole.