For a proof that the only ${\rm SO}(N)$ invariant tensors are products of $\delta_{ab}$'s and Levi-Civita symbols see M. Spivak, A Comprehensive Introduction to Differential Geometry (second edition) Vol. V, pp. 466-481. The number of pages required for the argument shows that it is not trivial.

For a proof that the only ${\rm SO}(N)$ invariant tensors are products of $\delta_{ab}$'s and Levi-Civita symbols see M. Spivak, A Comprehensive Introduction to Differential Geometry (second edition) Vol. V, pp. 466-481. The number of pages required for the argument shows that it is not trivial.

I've just looked up the third edition of Spivak vol V. What is needed is theorem 35 on page 327. This is the section entitled "A Smattering of Classical Invariant Theory." He writes in terms of scalar invariants, but of course an invariant tensor becomes a scalar invariant when contracted with enough vectors, and any such scalar invariant arises from an invariant tensor.