There is no way to export allocation history to Excel from the view allocation screen or elsewhere in the application.
As a workaround, use AOI to export Sales or Purchase Transactions with the Allocation User, Date, Value and Reference fields. This can help identify allocations, but is limited to transactions with no more than three allocations, as that is the field limit in the template.
If you need a report with specific allocation data, contact the support team to have a custom report created.