It is useful to consider vector spaces over C (complex numbers) rather than over R but in general can think of them as over a field F. The motivation fo the definition of a vector spaces comes from the properties of addition and scalar multiplication in Fn.
It is tempting to associate the elements of a vector space (over R say) with elements of Rn but it is important to remember that we don’t have a concept of distance in vector spaces (until we equip them with an inner product). As vector spaces alone (without an inner product), it is better to think of an n dimensional vector spaces as a combination of n different degrees of freedom.