Find a mapping from $R^{n \times n}$ to orthogonal matrices in the unit sphere such that distances do not increase.

Question

Let $O$ be the orthogonal group, let $B = \{A \in R^{n \times n} \text{ | } \lVert A \lVert \leq 1\}$.

Find a function $F:R^{n \times n} \rightarrow O \cap B$, such that $\forall_{A \in R^{n \times n}, Z \in B} \lVert F(A) - Z \lVert \leq \lVert A - Z \lVert$.

In other words, I would like to find a mapping from $R^{n \times n}$ to orthogonal matrices in the unit sphere such that distances do not increase.