In bivariate linear regression is there a direct relationship between sample size $n$, coefficient of determination $r^2$ and $\sigma_\beta$ (the standard error of coefficient $\beta$)? 

Assume data have been normalized so both target and predictor variable have $\sigma=1$.

Putting the question another way, does $\sigma_\beta$ tell me something different to $r^2$ or are they measures of the same thing?  Or, is it possible to have a strong, certain but unreliable link between two variables (large $\beta$, small $\sigma_\beta$, but small $r^2$)?

(In multivariate regression this doesn't apply as even with high $r^2$, $\sigma_\beta$ can indicate uncertainty as to which of the multiple predictors is causing the response).