Let $V$ and $W$ be vector spaces over the field $F$. We know that $V\otimes W$ is a vector space over $F$.
But how do we show closure under addition? For instance let $(a\otimes b), (c\otimes d)\in V\otimes W$. How do we know that $(a\otimes b)+(c\otimes d)\in V\otimes W$?
Since $V\otimes W$ is defined to be a vector space, it is closed under addition by definition. However, not every element of $V\otimes W$ is of the form $v\otimes w$. All we know from the definition is that there is a bilinear map $\mu:V\times W\to V\otimes W$ satisfying a certain universal property, and we write $\mu(v,w)=v\otimes w$. The map $\mu$ need not be surjective, and its image need not be closed under addition. So you can't necessarily write $a\otimes b+c\otimes d$ in the form $m\otimes n$. In general, all that you can say is that every element of $V\otimes W$ can be written as a sum $v_1\otimes w_1+v_2\otimes w_2+\dots+v_n\otimes w_n$.