Let's take the vector space $V$ of $R^2$, that is ... the set of all 2-uples.
My first question is : Can we represent any vector contained in $V$ as some linear combination of some basis in $V$?
I'm sure the answer is yes.
Following with that thinking, lets take the vectors that form the standard basis of $R^2$, that is the set containing the 2 orthonormal 2-uples.
They are clearly members of $R^2$. Since they are clearly members of $V$, they clearly can be represented as some linear combinational of some basis in $R^2$.
If we choose this basis the be the standard basis itself, dont we generate a recursive problem ?
If the standard basis vectors can be represented by the basis they form, how do we even know how to compose the basis vectors to generate the standard basis vectors if we first need the standard basis vectors ?