Let $V_k(\mathbb{R}^n)$ be the set of all orthonormal $k$-tuples of vectors $v_1,\ldots,v_k\in\mathbb{R}^n$. Let $M_{k,n}$ be the set of all $k\times n$ matrices. Let $W=\{A\in M_{k,n}\mid AA^t=I_k\}$, where $I_k$ is the identity $k\times k$ matrix. Prove that $V_k(\mathbb{R^n})$ can be identified with $W$.
(I'm not sure what "can be identified with" means. Maybe it just means showing there is a bijection.)
We just put in the vectors $v_1,\ldots,v_k$ as rows of $A$. Then these vectors are columns of $A_t$, so that $AA^t=I_k$ by definition of orthonormality. Conversely, given any $A$, we can let $v_1,\ldots,v_k$ be its rows.
Is there anything more to say about this?
It is exactly as you say. One says "identified" because you are not defining a bijection at random (after all, both sets have the same cardinality as $\mathbb R$), but rather you are creating your bijection in terms of the properties of the objects you are using.