You can create a free version of Azure SQL Database using PowerShell.
$sqlServer = New-AzureRmSqlServer -ServerName $sqlServerName -Location $desiredLocation.Location -ResourceGroupName $resourceGroup.ResourceGroupName -SqlAdministratorCredentials $sqlAdminCredential -ErrorAction Stop -Verbose
New-AzureRmSqlDatabase -DatabaseName $DatabaseName -ServerName $sqlServer.ServerName -ResourceGroupName $resourceGroup.ResourceGroupName -Edition 'Free' -RequestedServiceObjectiveName 'Free'
You can have only one free Azure SQL Server database per region, it reverts to Standard Edition after 365 days.
Here's the list of Azure locations that do support the creation of a limited time (365 days) Free Azure SQL Server and Azure SQL Database:
Location DisplayName SQL?
-------- ----------- -----
australiaeast Australia East WORKS
australiasoutheast Australia Southeast WORKS
centralindia Central India WORKS
eastasia East Asia WORKS
francecentral France Central WORKS
japaneast Japan East WORKS
japanwest Japan West WORKS
koreacentral Korea Central WORKS
koreasouth Korea South WORKS
northcentralus North Central US WORKS
northeurope North Europe WORKS
southeastasia Southeast Asia WORKS
southindia South India WORKS
uksouth UK South WORKS
ukwest UK West WORKS
westcentralus West Central US WORKS
westeurope West Europe WORKS
westindia West India WORKS
westus2 West US 2 WORKS
Hope this helps.