If you're doing an OD Cost Matrix analysis in ArcGIS, then it's not possible to retrieve the network path between each origin and destination. The OD Cost Matrix solver uses the road network when calculating the travel times and distances, but it optimizes performance and memory usage by not storing those network paths. It only stores the calculated costs, and it can optionally generate a straight line between each origin and destination.
If you want to get the geometry of the on-street routes between origins and destinations, you can use the Closest Facility solver instead. The functionality is very similar to OD Cost Matrix, but it does have the option to store the geometry (and it's a little slower). If you want to connect all origins to all destinations, you just need to set the number of facilities to find for the "closest" facility to a number >= your number of destinations.
Here's the Closest Facility tutorial: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/closest-facility-tutorial.htm
