Using rfc5545 is there a way to represent an event that starts on the first day of each month and ends on the last day of each month and repeats every month?
It's slightly different than a daily repeating event which is not ideal for my use case.
short answer: No
This is driven by the fact that the RFC5545 clearly states that an event cannot have a month duration, only days, weeks or seconds.
3.3.6. Duration
[...] Note that unlike [ISO.8601.2004], this value type doesn't support the "Y" and "M" designators to specify durations in terms of years and months.
(emphasis mine)