We show that the localized induction approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.