Standard error notation Dealing with a standard error of a mean 
$$ \widehat{SE}_{\bar x} = \hat \sigma_{\bar x} = \frac{s_x}{\sqrt{n}}$$ 
$$ SE_{\bar x} = \sigma_{\bar x} = \frac{\sigma_x}{\sqrt{n}}$$ 
Is this the standard correct notation?
 A: I am not aware of any strict notational convention for the standard error, other than the fact that it is usually denoted either by $se$ or $SE$, often with a subscript denoting the estimator of concern.  Your question uses a subscript $\bar{x}$ to denote the sample mean (as an estimator for the mean parameter) and one could quibble in this case with your use of lower case notation, which usually denotes an estimate (outcome), rather than estimator (random variable).  If it were me, I would prefer:
$$\begin{array}\ 
\text{True standard error} & & & 
SE_\bar{X} \equiv \mathbb{S}(\bar{X}) = \frac{\sigma_X}{\sqrt{n}}, \\[6pt]
\text{Estimated standard error} & & & 
\widehat{SE}_\bar{X} \equiv \hat{\mathbb{S}}(\bar{X}) = \frac{s_X}{\sqrt{n}}. \\[6pt]
\end{array}$$
However, this point is minor, since the use of $\bar{x}$ can reasonably denote the estimator (so long as that is not inconsistent with its use in the rest of your work), and in any case, we know what you mean.  I do not think any reader would have difficulty with your chosen notation.
