Cybersecurity education and training are being conducted on an ever-increasing scale, as most organizations need to improve their readiness in dealing with the more and more frequent cyberattacks. However, most systems used for such education and training purposes are built from scratch, are highly customized, and often proprietary. This is true especially for complex activities that include hands-on practice, such as Capture The Flag (CTF) competitions and realistic cyber range training. Moreover, the specificities of these platforms create an important overhead, both for instructors, who need to develop training content and learn how to use them, and also for trainees, who need to each time adjust to a different platform. In this paper, we present our approach of integrating cybersecurity training activities, both for technical and awareness training, with Learning Management Systems (LMSs). In particular, our system--named CyLMS--provides integration from content point of view with most LMSs through the use of the SCORM format for packaging the training content. Moreover, additional CyLMS modules make possible a tighter integration with the Moodle LMS, a widely-used e-learning platform, for tasks such as automatic activity management and hands-on environment access. In this way, both instructors and trainees benefit from standard interfaces for checking the training content, answering questions, managing the results, etc. The paper includes an evaluation of CyLMS from a functionality, user and performance perspectives that demonstrates its applicability to actual training activities. While so far we have only used CyLMS in the cybersecurity context, the platform is sufficiently generic to be applied to other education activities, as a learning content management tool that facilitates training content creation and sharing.