Suppose that we defined a tensor product of vector spaces $U$ and $V$ as a quotient of a vector space with basis $V \times W$ by the vector space spanned by $-(u_1+u_2,v)+(u_1,v)+(u_2,v), -(u,v_1+v_2)+(u,v_1)+(u,v_2),-(au,v)+(u,av)$.
I have proved the universal property for two vector spaces. But how then extend it to a tensor product of $n$ vector spaces?
You can define it recursively by $V_1 \otimes \cdots \otimes V_n = \left( V_1 \otimes \cdots \otimes V_{n-1} \right) \otimes V_n$, but you want to check associativity.
See this answer.