Hmm, I always disliked coordinate approach to (multi)linear algebra. To me, the notion of a tensor product clicked with the following definition:

Tensor product v₁⊗v₂ of vectors form V is bilinear form on the dual space V* defined by v₁⊗v₂(ω₁,ω₂) = v₁(ω₁)v₂(ω₂). Tensor product V⊗V is space of all bilinear forms on V*,

This can easily be generalized to products of forms (covariant tensors), and mixed tensor products. Getting coordinate representation in basis (E₁ ... Eₙ) is trivial: (v₁⊗v₂)ₖ,ₗ = v₁⊗v₂(εₖ,εₗ) where εₖ is dual to Eₖ...