need "gauss like" PDF with skewness

loloPF

I am looking for a Probability Density Function that has the following properties:

1. is defined on R like the gaussian
2. has a non null (and non constant) skewness that is controlled by a parameter
3. degenerates towards the gaussian

At the moment I am using a beta function but it does not meet requirements (1) and (3).

I checked the Pearson system but only the generating differential equation degenerates to that of the gaussian, not its solution... see what I mean?

EnumaElish

I say, good luck to you. I don't think any of the commonly used distributions satisfy all of the 3 properties.