The problems of the classical dynamics of a relativistic string are intimately related to the theory of two-dimensional extremal surfaces in $n$-dimensional pseudo-Euclidean space $E^1_n$. In three-dimensional space-time $E^1_3$, it is possible to exploit fully the formalism of the Gaussian theory of two-dimensional surfaces, the surface being specified to within shifts by its first and second quadratic forms. Integration of the derivation formulas for the basic vectors $\partial x_\mu(\tau,\sigma)/\partial\tau=\dot x_\mu(\tau,\sigma)$, $\partial x_\mu(\tau,\sigma)/\partial\sigma=x_\mu'(\tau,\sigma)$ are the tangent vectors to the surface and $m_\mu(\tau,\sigma)$ is the normal to the surface at the given point $\tau,\sigma$) yields a representation for these vectors in a natural basis satisfying the orthonormal gauge $(\dot x_\mu\pm x'_\mu)^2=0$ and d'Alembert's equation $\ddot x_\mu(\tau,\sigma)-x''_\mu(\tau,\sigma)=0$ in the string dynamics. This representation can be generalized to a pseudo-Euclidean space $E^1_n$, of any dimension $n$. For a relativistic string in $E^1_n$ a representation is obtained that contains $n-2$ arbitrary functions and satisfies the gauge conditions, the equations of motion, and the boundary conditions for a free string.