The Riemannian submersion π : SO_0(1,n) ->H^n is a principal bundle and its fiber at π (e) is the imbedding of SO(n) into SO_0(1,n) , where e is the identity of both SO_0(1,n) and SO(n). In this study, we associate a curve, starting from the identity, in SO(n) to a given piecewise smooth surface with boundary, homeomorphic to the closed disk D^2, in H^n such that the starting point and the ending point of the curve agree with those of the horizontal lifting of the boundary curve of the given surface with boundary, respectively, and that the length of the curve is as same as the area of the given surface with boundary. In addition, the curve in SO(n) relates the connection of its tangent vector to the curvature of some point in SO_0(1,n).