Count the sum of the determinants

To each positive integer with n^2 number of digits, we can associate a corresponding determinant by writing the digits in order across the rows.

That is for n = 2, we have n^2 = 4. Thus take all 4-digit positive integers. For 7615, we have the determinant
| 7 6 |
| 1 5 |
which is equal to 35 - 6 = 29.

Can you find the sum of all determinants associated with n^2-digit integers as a function of n?

Just as a motivating example - for n = 1, we have only 9 determinants and their sum is 45. For n = 2, we have 9000 determinants.