can the ROW_NUMBER() function be used without the PARTITION clause?

Partition By clause is not mandatory.

select *, Row_number() Over (order by salary desc) rn from emp-- this query just assign sequential number after the data sorted on sal column in descending order...

Gender-wise Salary rankings:

select *, Row_number() Over (partition by gender order by salary desc) rn from emp-- this query assigns ranks once after data partioned on gender column

It requires an order by, where you can specify any columns, for example:

select *, row_number() over(order by MyColumn1) as num 
from MyTable

If the order is not important, then, for example:

select *, row_number() over(order by @@version) as num 
from MyTable

To assign numbers in random order:

select *, row_number() over(order by newid()) as num 
from MyTable