When packet gets converted into a frame, it gets a "Destination hardware address", "Source Hardware address", "Ether-type field" or "length", "data field" and a "Frame check sequence field".
Now, at this particular point, if the destined host is on the local network, then a request for Hardware address of the local host via an ARP is made by IP at network layer, and once it is received, the hardware address is handed down to data link layer so that it can convert it into a frame and perform trustworthy transmission over physical link.
If, however, the packet is destined for a remote host, then how would it get the hardware address of the default gateway (or next router)? because each time a packet is sent between routers , it is converted into frames by Data Link Layer . and stripped-off to the packet at receiving router. so, from where the Data Link Layer will get the hardware address of the Default Gateway (router) to fill the "destination Hardware address" field as it did in the case of local host?