If $v$ is an element of a vector space $V$ and for example $\mathcal{B}=\{e_1,e_2,e_3\}$ is a basis of $V$, then, at least, there should be another basis for $V$ in which the vectors of $\mathcal{B}$ can be expressed, but at the same time, the vectors of this other basis must also be expressed using $\mathcal{B}$'s vectors.
Why is there no problem here?
No, there is no problem, since a vector does not need a basis to be expressed. Take, for instance, the vector space $V$ of all continuous functions from $[0,1]$ into $\mathbb R$. And now conside the map $f\colon[0,1]\longrightarrow\mathbb R$ defined by $f(x)=e^x$. It belongs to $V$, right? But I had no need of a basis of $V$ in order to express it.