Special cases of distributions I'm confused on the special cases for some distributions and i'm looking for some clarity. 
I know that the standard normal distribution is a special case of the normal distribution with a mean of 0, standard deviation of 1 and that the skewness and kurtosis of any normal distribution is 0 and 3 respectively. 
What i'm looking for is this case for these distributions:
T distribution, Skewed-t distribution, skewed normal distribution.
By 'this case' I don't mean the standard normal distribution for these distributions. I'm just looking for what would be the equivalent special case relative to the distribution.
Edit: Another question I have is specifically for the student-t distribution. What would be the specific case for it and what values would its parameters have?
 A: I think the reason you are confused is that there really is no equivalent.
The "standard normal" is used and is useful for two reasons. First, the normal is very very common.  Second, in the old days when we had to look up values in tables, we needed to make the Normal distribution have one table. So, they picked a mean of 0 and sd of 1, somewhat arbitrarily, and people converted other normals to that. 
The t distribution is, in a way, automatically standardized (the unstandardized version is the noncentral t distribution).  That is, it doesn't have parameters for mean and variance, only for degrees of freedom. There is no way to make a equivalent standard for degrees of freedom. The t is the special case, but it is a whole bunch of cases, because df varies.
The skew normal has 3 parameters (for shape, scale and location) and the standard normal is a special case of this distribution. 
A: I disagree somewhat with Peter's characterization of the situation with the t-distribution, so I'll outline where I'd put it differently  (this started as a comment but got too long with all the links).
The t-distribution that's in t-tables (e.g. see the wikipedia page on the distribution) - that you'd use for tests and CIs and so on - is "standard" in the sense that is has location parameter 0 and scale parameter 1. (The variables used for those calculations get standardized.)
You can also have [a location-scale family]((e.g. see the wikipedia page on the distribution for the t-distribution. That corresponds to the general normal (not that for the t-distribution, the scale parameter $\sigma$ is not the standard deviation, however). This is the version that's needed when modelling data.
[You might find the Wikipedia article on location-scale families useful background]

Moments of the t-distribution; this information is readily obtained from the wikipedia article on it:
The mean and variance of the t-distribution in terms of the parameters are given by $\mu$ and $\sigma^2 \frac{\nu}{\nu-2}$; the skewness and kurtosis are $0$ and $3+\frac{6}{\nu-4}$.
However, matching the sample kurtosis is not generally a suitable way to choose a $t$-distribution for modelling data. 
Further, simply comparing sample skewness and kurtosis to population skewness and kurtosis -- no matter how close they are -- doesn't tell you that you have a suitable model. e.g. even if your sample skewness is 0 and sample kurtosis is 3, it doesn't imply a normal distribution is a good model, and it doesn't imply that assuming a normal distribution won't lead to any problems; the issue can be even worse with the $t$ (where the sample kurtosis in particular can be quite  biased and very noisy)
