Denote $e_1 = (1,0,0), e_2 = (0,1,0), e_3 = (0,0,1)$.
You already noted that the vectors $e_i \otimes e_j$ form a basis of $V \otimes V$. It follows that the vectors $e_i \otimes e_j + Sym^2(V)$ form a spanning set of the quotient space $(V \otimes V)/ Sym^2(V)$. It follows that any maixmal linearly independent subset of those vectors will be a basis for our quotient space.
Now, consider the vectors
e_1 \otimes e_2 + Sym^2(V), \quad e_1 \otimes e_3 + Sym^2(V), \quad
e_2 \otimes e_3 + Sym^2(V), \quad
Why do these vectors form a maximal linearly independent set? That is, why does adding any vector $e_i \otimes e_j + Sym^2(V)$ make our set linearly dependent? Once you answer that, you can know with certainty (by the "dimension theorem" of linear algebra) that this set is a basis.
I'll leave it to you to work through the addition table.