Prove that tensor of the type $(1,1)$ which is invariant under orthogonal transformations of $\mathbb{R}^n$ is proportional to the tensor $\delta^{i}_{j}$.

Approach: Let $V=\mathbb{R}^n$ and $T:V\times V^*\to \mathbb{k}$ be a tensor of type $(1,1)$. Let $\{e_1,\dots,e_n\}$ and $\{\tilde{e}_1,\dots,\tilde{e}_n\}$ be two basis of $V$ such that matrix $C$ is transformation matrix from $(e)$ to $(\tilde{e})$. Then $\tilde{e}_j=c^l_je_l$ and $\tilde\varepsilon^i=d^i_k\varepsilon^k$ where by $\{\varepsilon^i\}$ and $\{\tilde{\varepsilon}^i\}$ I mean corresponding dual basis.

Let $\tilde{T}^i_j=T(\tilde{e}_j,\tilde{\varepsilon}^i)=T(c^l_je_l,d^i_k\varepsilon^k)=c^l_jd^i_kT(e_l,\varepsilon^k)=c^l_jd^i_kT^k_l$, where $c^i_j$ and $d^i_j$ are elements of $C$ and $C^{-1}$, respectively.

I was wondering what does mean that our tensor is invariant under orthogonal transformation?

I have the following hypothesis: maybe $T^i_j=c^l_jd^i_kT^k_l$ for any orthogonal matrix $C$?

Note that in the LHS I wrote $T^i_j$ instead of $\tilde{T}^i_j$.

Am I wrong? Anyway what is the definition of invariant tensor? Would be very grateful for any comments!