Yes. To do this, enter your own details as the recipient on the product page. No name is written on the gift card itself, so the recipient details are only used for delivery.
However, please note that for email and text message deliveries, the email address or phone number to which the gift card was originally sent must be used for verification when registering your Global Gift Card or downloading your Partner Gift Card. This only applies if you forward the delivery email or SMS.
You can also download the actual gift card PDF and give that instead. In that case, no further verification is needed.