In fact this counting scheme is the proof that there is no way to express the general rank 2 tensor ##T## as ##T=V\otimes W##. ##V## and ##W## between them have ##2n## components, while a tensor of rank 2 has ##n^2## components. Therefore, there is not enough freedom in 2 vectors to specify a general rank 2 tensor.Let's use the dyadic terminology that here might clarify things:a general second order tensor is called a dyadic. And certainly it doesn't have to be result of the product of two vectors. A second order tensor that is the direct product of two vectors is a dyad: ad dyadic tensor of rank one, using the term rank with a different meaning than in previous posts where it was synonimous of order . Now every dyadic can be expressed as a linear combination of dyads.
Now the stress-energy tensor is not a dyad so I can see it is not good for my purpose, but not for the reason you mention as in general the dyadic product of two vectors has ##n^2 ## independent components (with n being the dimension), that may or not be reduced in function of the symmetries of the resultant tensor.
Matterwave, it would be interesting if you could provide some reference that shows that the number of independent components after the tensor product operation is performed on two vectors is the sum n+n instead of ##n^2##, I find it difficult since each component of a dyadic product is obtained by multiplication of the components of each vector and except some of the vector components are zero one gets ##n^2## independent components before possible reductions due to symmetry.
Note that there is something that sometimes is referred to simply as dyad, that is a combination of two n-vectors juxtaposed that doesn't involve any multiplication, that is simply a linear transformation matrix and that indeed has just n+n independent components at most.
If you want a source, you can look at Bernard Schutz Geometrical methods of mathematical physics. Chapter 2, exercise 2.5 actually asks the student to prove this themselves (for the solution, see the appendix, wherein he says exactly what I said).