The Cortana Skills Kit was design to allow developers to make use of services and/or bots created with the Microsoft Bot Framework and publish them to Cortana as a new skill. Microsoft officials also said that developers using the kit could opt to repurpose code from existing Alexa skills.
Microsoft also announced this week a couple of new features for developers and customers of Cortana skills. Microsoft added support for Skills suggestions and Cortana skills chaining -- both of which Microsoft is touting as making use of its advances in search and AI.
Microsoft also announced this week that developers can take advantage of the Microsoft Knowledge Graph Exchange to more easily build skills using prebuilt modules, called botlets, that encapsulate data, language understanding, and logic.
Amazon and Microsoft announced a partnership deal involving Alexa and Cortana this past Summer, the first deliverables of which were slated to arrive before year-end. I asked Microsoft officials today if Alexa and Cortana would be able to work together as announced before the year is out. So far, no word back from Microsoft.