An extended Courant element is constructed on an $n$ dimensional polytope $K,$ which reduces to the usual Courant element when $K$ is a simplex. The set of the degrees of freedom consists of function values at all vertices of $K,$ while the shape function space $P_K$ is formed by repeatedly using the harmonic extension from lower dimensional face to higher dimensional face. Several fundamental estimates are derived on this element under reasonable geometric assumptions. Moreover, the weak maximum principle holds for any function in $P_K,$ which enables us to use the element for approximating an obstacle problem in three dimensions. The corresponding optimal error estimate in $H^1$-norm is also established. Numerical results are reported to verify theoretical findings.