  Let $\Gamma_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let $ Z_n$
be the $p\times q$ upper-left submatrix of $\Gamma_n,$ where $p=p_n$ and
$q=q_n$ are two positive integers. Let $G_n$ be a $p\times q$ matrix whose $pq$
entries are independent standard normals. In this paper we consider the
distance between $\sqrt{n} Z_n$ and $ G_n$ in terms of the total variation
distance, the Kullback-Leibler distance, the Hellinger distance and the
Euclidean distance. We prove that each of the first three distances goes to
zero as long as $pq/n$ goes to zero, and not so this rate is sharp in the sense
that each distance does not go to zero if $(p, q)$ sits on the curve $pq=\sigma
n$, where $\sigma$ is a constant. However, it is different for the Euclidean
distance, which goes to zero provided $pq^2/n$ goes to zero, and not so if
$(p,q)$ sits on the curve $pq^2=\sigma n.$ A previous work by Jiang
\cite{Jiang06} shows that the total variation distance goes to zero if both
$p/\sqrt{n}$ and $q/\sqrt{n}$ go to zero, and it is not true provided
$p=c\sqrt{n}$ and $q=d\sqrt{n}$ with $c$ and $d$ being constants. One of the
above results confirms a conjecture that the total variation distance goes to
zero as long as $pq/n\to 0$ and the distance does not go to zero if $pq=\sigma
n$ for some constant $\sigma$.
