Dimension of $GL(n, \mathbb{R})$ Why is the group $GL(n, \mathbb{R})$ of dimension $n^{2}$?

I'm sorry people misunderstood your question. You are clearly asking about dimension in terms of linear algebra and not in terms of manifolds.
The set $GL(n)$ is not a subspace. To see this, take any invertible matrix $A$, then $-A$ is also invertible. But $A-A=0$ is obviously not invertible, thus $GL(n)$ is not a vector subspace for any $n$ so it makes no sense to talk about it's dimension in terms of independent vectors.
However, GL(n) is what is known as a submanifold which means that although $GL(n)$ is not a vector space, you can "locally" (i.e. around some neighborhood around the set) view it a vector space structure. But this is something you shouldn't worry about in linear algebra.
