Should I get an area rug for this space or not?

A square or rectangular rug would work picking up the various colors found in the curtains and pillows along with the sofa colors. A rug will soften the look and add a punch of color to the floor.

I would go Bear Skin. Beautiful floor BTW.

Hello Ang! An ra rug would look wonderful in this space! it would pull the space together nicely, define it a bit more, and make it feel more cozy. You could pull a color from one of the patterned pillows.

Something like this?

Beautiful floors! Room needs a rug to define the space. Wall also needs artwork of some sort. You have a lot of solid neutrals going on here, choose something with color for both the artwork and rug. Here is how to determine what size rug to purchase.

https://www.rugandhome.com/blogs/decorating-guide/17369272-rugs-101-selecting-a-rug-size

You can use a rug during colder months and roll it up for the summer. Rugs just feel warmer, but hard floors in the summer seem cooler. You can that rug in whenever your in the moodtoo

Our living room has a similar setup and the addition of a rug made such a big difference! Here are two diagrams that show you the best ways to place furniture depending on the size of the rug you choose. Basically, you want all of your furniture to at least partially fit on top of the rug.

https://www.pinterest.ca/pin/AZDz-56Y4fbdKKnCq3kc18qKN09TQHNXDQGPEL7aKeJpW4kARyh4R5Y/

https://www.pinterest.ca/pin/231442868327127583/

Your floors are beautiful, it looks like you have little ones, it may be hard to keep an area rug clean and vacuumed although I agree it would be beautiful, maybe a possibility to put washable scatter rugs in that area.