I'm pretty sure this is the first time I've ever tried to answer another person's questions about anything tensor-related, so I apologize if I get something (or everything) totally wrong.

@Igor I imagine the wikipedia article is saying that \$$e_i\$$ and \$$f_j\$$ are basis vectors to two vectors spaces \$$E\$$ and \$$F\$$, which you then want to tensor together to get \$$E \otimes F\$$. It sounds like you're then asking how to find the basis vectors of \$$E \otimes F\$$.

Suppose both \$$E\$$ and \$$F\$$ are two-dimensional; I'll call their basis vectors \$$e_1, e_2, f_1, f_2\$$.

To write down all the \$$e_i \otimes f_j\$$, you just combine every way you can (that fits the pattern):

\$$e_1 \otimes f_1\$$

\$$e_1 \otimes f_2\$$

\$$e_2 \otimes f_1\$$

\$$e_2 \otimes f_2\$$

Those are your four basis vectors of \$$E \otimes F\$$. (Recall that we expected 4, because the dimension of \$$E \otimes F\$$, which is itself a vector space, is dim(E) * dim(F) = 2*2 = 4.)