As the title says, here's my "proof":
Let U be some orthogonal matrix:
Uᵀ = U⁻¹
∴ U Uᵀ = Uᵀ U = I
Considering the ijth element:
(U Uᵀ)ᵢⱼ = (Uᵀ U)ᵢⱼ = δᵢⱼ
∑UᵢₖUᵀₖⱼ = ∑UᵀᵢₖUₖⱼ = δᵢⱼ
∑UᵢₖUⱼₖ = ∑UₖᵢUₖⱼ = δᵢⱼ
The scalar product between two vectors is given by ⟨a,b⟩ = ∑aᵢ*bᵢ
We want to get ⟨Uⱼ,Uᵢ⟩ = ⟨Uⁱ,Uʲ⟩ = δᵢⱼ for orthonormality to be satisfied in the rows and columns of U.
But this isn't what I have, I'm only able to get the complex conjugate for when U is real (unitary) but not for the general case of orthogonal maps.