Lie algebra of complex Lie group equal to complexification of a Lie algebra?

Question

For example, let $G=GL_N(\mathbb{R})$ and $G_{\mathbb{C}}= GL_N(\mathbb{C})$.

Let $g_{\mathbb{R}}$ and $g_{\mathbb{C}}$ be their corresponding real and complex Lie algebra.

Then I am wondering whether $g_{\mathbb{C}}$ is the complexification of $g_{\mathbb{R}}$. (i.e. $g_{\mathbb{C}}= g_{\mathbb{R}} \otimes_{\mathbb{R}} \mathbb{C}$?)

Thank you very much in advance!