You can inspect gradients by e.g. using register_hook on all parameters and then by printing or storing the value or alternatively by checking the .grad attribute of all parameters after the backward call:
loss.backward()
for name, param in model.named_parameters():
    print(name, param.grad)

To see if the gradient is vanishing you could print its norm or absolute value (or any other reduction which makes sense) and compare the magnitude with and without the skip connections.