  An $n \times n$ matrix with $\pm 1$ entries which acts on $\mathbb{R}^n$ as a
scaled isometry is called Hadamard. Such matrices exist in some, but not all
dimensions. Combining number-theoretic and probabilistic tools we construct
matrices with $\pm 1$ entries which act as approximate scaled isometries in
$\mathbb{R}^n$ for all $n$. More precisely, the matrices we construct have
condition numbers bounded by a constant independent of $n$.
  Using this construction, we establish a phase transition for the probability
that a random frame contains a Riesz basis. Namely, we show that a random frame
in $\mathbb{R}^n$ formed by $N$ vectors with independent identically
distributed coordinates having a non-degenerate symmetric distribution contains
many Riesz bases with high probability provided that $N \ge \exp(Cn)$. On the
other hand, we prove that if the entries are subgaussian, then a random frame
fails to contain a Riesz basis with probability close to $1$ whenever $N \le
\exp(cn)$, where $c<C$ are constants depending on the distribution of the
entries.
