Let $\mathbb{R}^{N \times 1}$ be a D6081: Standard inner product space of real column matrices.
Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix such that

(i)	\begin{equation} A = \begin{bmatrix} A_1 & A_2 & \cdots & A_N \end{bmatrix} \end{equation}

Then $A$ is an orthonormal real matrix if and only if $A_1, \ldots, A_N$ is an D10: Orthonormal collection in $\mathbb{R}^{N \times 1}$.