I'm not sure if this answers your question but in order for a set of vectors to be a basis of a vector space it has to be linear independant and has to span the entire space. Now if you're given a set of n linearly independant vectors in you only have to show that it spans . But as you said yourself this is indeed the case since you can express any vector as a linear combination of vectors from your set.
If that isn't satisfactory, you could also think about it this way: Suppose you have n linearly independant vectors and another vector w not in the span of your . Then is linear dependant since it's a set of n+1 vectors in a vector space of dimension n. This means there are not all 0 with . Now because otherwhise would be linear dependant. Therefore but this means w is in the span of a contradiction.
Not exactly what you’re looking for?