To follow-up @Vadim's comment, I thought I would add how OLS is represented in matrix form. This provides a visual representation for you, in which you can see that the intercept in the OLS model is indeed represented by a vector of unities (i.e. ones). So essentially, the intercept is calculated much like other $b$ parameters, but the vector of 1s is used to calculate the intercept.
Then you use the usual formula to calculate model parameters:
Then remember: the standard error of each parameter is the standard deviation of each parameter's sampling distribution. Hence you may proceed to calculate variance-covariance matrix of your parameters including the intercept. This can be done in the following way. Formulae included per your request.
Final note: Apologies for styling, I did not have much time to use LaTeX for formula styling, so included pictures instead.