  We prove that the vast majority of symmetric states of qubits can be
decomposed in a unique way into a superposition of spin 1/2 coherent states.
For the case of two qubits, the proposed decomposition reproduces the Schmidt
decomposition and therefore, in the case of a higher number of qubits, can be
considered as its generalization. We analyze the geometrical aspects of the
proposed representation and its invariant properties under the action of local
unitary and local invertible transformations. As an application, we identify
the most general classes of entanglement and representative states for any
number of qubits in a symmetric state.
