What is $\mathfrak{sl}(2,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}$?

Question

I am trying to understand Lie algebras and I've come across this $$\mathfrak{sl}(2,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}\cong\mathfrak{sl}(2,\mathbb{C})\cong\mathfrak{so}(3,\mathbb{C}),$$ but I don't really know how the first object works; does it just consist of tensoring each matrix entry by $i=\sqrt{-1}$? Or is it somewhat more complex?