It would be easier to answer this question if we could discuss some specific soft skill. Let’s take leadership for example.
Leadership is a different thing for different stakeholders, so the first obvious step is to define who is interested in measuring the leadership and the second question is why this is important.
For example, we can answer that:
1. The stakeholder is the manager of the company, who needs to measure the leadership skills of key team members
2. The purpose of the measurement (answer to the 2nd question) might be, for example, the need to select the best person to lead the team (why not?!)
With this context in mind we can find an appropriate measurement tool. For example, we can setup some role-playing game where leadership skills will be visible. But be careful with this, as not all leaders like playing games. And you need to measure leadership, but not just the ability to play games.
A good option would be to do several observations/experiments in a controlled environment. An imagination will help to come up with the ideas about how to prepare those experiments.
As you can see it is not a classical approach to a KPI, but is still very effective. Once you see some results, you can quantify them and pack them into KPIs if needed.